From cc452130d65a930deec37633bea57aba460cf477 Mon Sep 17 00:00:00 2001
From: m-iancu <mihnea.iancu88@gmail.com>
Date: Fri, 1 Jul 2016 14:18:59 +0200
Subject: [PATCH] new version of mws harvests
---
export/mwscrawler/1.html | 38 +-
export/mwscrawler/10.html | 18 +
export/mwscrawler/11.html | 532 +++++++++++++++++++++++
export/mwscrawler/12.html | 349 +++++++++++++++
export/mwscrawler/13.html | 195 +++++++++
export/mwscrawler/14.html | 888 ++++++++++++++++++++++++++++++++++++++
export/mwscrawler/15.html | 202 +++++++++
export/mwscrawler/16.html | 78 ++++
export/mwscrawler/17.html | 19 +
export/mwscrawler/18.html | 7 +
export/mwscrawler/19.html | 7 +
export/mwscrawler/2.html | 36 +-
export/mwscrawler/20.html | 14 +
export/mwscrawler/21.html | 7 +
export/mwscrawler/22.html | 58 +++
export/mwscrawler/23.html | 14 +
export/mwscrawler/24.html | 29 ++
export/mwscrawler/25.html | 7 +
export/mwscrawler/26.html | 73 ++++
export/mwscrawler/3.html | 300 +------------
export/mwscrawler/4.html | 340 +++++++++++++--
export/mwscrawler/5.html | 66 ++-
export/mwscrawler/6.html | 23 +
export/mwscrawler/7.html | 7 +
export/mwscrawler/8.html | 41 ++
export/mwscrawler/9.html | 7 +
26 files changed, 2958 insertions(+), 397 deletions(-)
create mode 100644 export/mwscrawler/10.html
create mode 100644 export/mwscrawler/11.html
create mode 100644 export/mwscrawler/12.html
create mode 100644 export/mwscrawler/13.html
create mode 100644 export/mwscrawler/14.html
create mode 100644 export/mwscrawler/15.html
create mode 100644 export/mwscrawler/16.html
create mode 100644 export/mwscrawler/17.html
create mode 100644 export/mwscrawler/18.html
create mode 100644 export/mwscrawler/19.html
create mode 100644 export/mwscrawler/20.html
create mode 100644 export/mwscrawler/21.html
create mode 100644 export/mwscrawler/22.html
create mode 100644 export/mwscrawler/23.html
create mode 100644 export/mwscrawler/24.html
create mode 100644 export/mwscrawler/25.html
create mode 100644 export/mwscrawler/26.html
create mode 100644 export/mwscrawler/6.html
create mode 100644 export/mwscrawler/7.html
create mode 100644 export/mwscrawler/8.html
create mode 100644 export/mwscrawler/9.html
diff --git a/export/mwscrawler/1.html b/export/mwscrawler/1.html
index eb6bd2b..6c78dae 100644
--- a/export/mwscrawler/1.html
+++ b/export/mwscrawler/1.html
@@ -1,37 +1,9 @@
<html>
<head>
- <title>TODO</title>
- <meta name="url" content="http://www.gap-system.org/Manuals/doc/ref/TODO#X7F10E951789D6EDF"></meta>
+ <title>2.5 Entities and Special Characters</title>
+ <meta name="url" content="http://www.gap-system.org/Manuals/doc/ref/TODO#X83A355E68485D6D1"></meta>
</head>
- <body> <h4 xmlns="http://www.w3.org/1999/xhtml">2.4 <span class="Heading">Lists and Tables</span></h4><p xmlns="http://www.w3.org/1999/xhtml">[→ <a shape="rect" href="chapB_mj.html#X7BB822947F626E1A"><span class="RefLink">B.9</span></a>]</p><p xmlns="http://www.w3.org/1999/xhtml">There are</p><ul xmlns="http://www.w3.org/1999/xhtml"><li><p>lists</p>
-
-</li><li><p>enumerations, and</p>
-
-</li><li><p>tables</p>
-
-</li></ul><p xmlns="http://www.w3.org/1999/xhtml">or:</p><ol xmlns="http://www.w3.org/1999/xhtml"><li><p>lists</p>
-
-</li><li><p>enumerations, and</p>
-
-</li><li><p>tables</p>
-
-</li></ol><p xmlns="http://www.w3.org/1999/xhtml">or with marks:</p><dl xmlns="http://www.w3.org/1999/xhtml"><dt><strong class="Mark">lists:</strong></dt><dd><p>not numbered</p>
-
-</dd><dt><strong class="Mark">enumerations:</strong></dt><dd><p>numbered</p>
-
-</dd><dt><strong class="Mark">tables:</strong></dt><dd><p>two-dimensional</p>
-
-</dd></dl><p xmlns="http://www.w3.org/1999/xhtml">Lists can also be nested:</p><ol xmlns="http://www.w3.org/1999/xhtml"><li><ol><li><p>first item of inner enumeration</p>
-
-</li><li><p>second item of inner enumeration</p>
-
-</li></ol>
-</li><li>
-<ul><li><p>first item of inner list</p>
-
-</li><li><p>second item of inner list</p>
-
-</li></ul>
-</li></ol><p xmlns="http://www.w3.org/1999/xhtml">Here is a <em>table</em>:</p><div class="pcenter" xmlns="http://www.w3.org/1999/xhtml"><table class="GAPDocTable"><caption class="GAPDocTable"><b>Table: </b>Prices</caption><tbody><tr><td colspan="1" rowspan="1" class="tdright">Object</td><td colspan="1" rowspan="1" class="tdcenter">Price</td><td colspan="1" rowspan="1" class="tdleft">available</td></tr><tr><td colspan="1" rowspan="1" class="tdright">Shoe</td><td colspan="1" rowspan="1" class="tdcenter">$1,00</td><td colspan="1" rowspan="1" class="tdleft">there</td></tr><tr><td colspan="1" rowspan="1" class="tdright">Hat</td><td colspan="1" rowspan="1" class="tdcenter">$2,00</td><td colspan="1" rowspan="1" class="tdleft">not there</td></tr></tbody></table><br/><p>Â </p><br/>
-</div> </body>
+ <body> <h4 xmlns="http://www.w3.org/1999/xhtml">2.5 <span class="Heading">Entities and Special Characters</span></h4><p xmlns="http://www.w3.org/1999/xhtml">[→ <a shape="rect" href="chapB_mj.html#X80B478CD7E584F6F"><span class="RefLink">B.10</span></a>]</p><p xmlns="http://www.w3.org/1999/xhtml">Here is a table of special characters, the first two are special for XML and must be typed in by entities in <strong class="pkg">GAPDoc</strong> documents. The other characters are special for LaTeX but in <strong class="pkg">GAPDoc</strong> they can be typed directly.</p><div class="pcenter" xmlns="http://www.w3.org/1999/xhtml"><table class="GAPDocTable"><caption class="GAPDocTable"><b>Table: </b>Special characters in character data</caption><tbody><tr><td colspan="1" rowspan="1" class="tdcenter"><code class="code">&</code></td><td colspan="1" rowspan="1" class="tdcenter"><code class="code"><</code></td><td colspan="1" rowspan="1" class="tdcenter"><code class="code">></code></td><td colspan="1" rowspan="1" class="tdcenter"><code class="code">#</code></td><td colspan="1" rowspan="1" class="tdcenter"><code class="code">$</code></td><td colspan="1" rowspan="1" class="tdcenter"><code class="code">%</code></td><td colspan="1" rowspan="1" class="tdcenter"><code class="code">~</code></td><td colspan="1" rowspan="1" class="tdcenter"><code class="code">\</code></td><td colspan="1" rowspan="1" class="tdcenter"><code class="code">{</code></td><td colspan="1" rowspan="1" class="tdcenter"><code class="code">}</code></td><td colspan="1" rowspan="1" class="tdcenter"><code class="code">_</code></td><td colspan="1" rowspan="1" class="tdcenter"><code class="code">^</code></td><td colspan="1" rowspan="1" class="tdcenter"><code class="code"> </code></td></tr></tbody></table><br/><p> </p><br/>
+</div><p xmlns="http://www.w3.org/1999/xhtml">And here are the predefined entities in <strong class="pkg">GAPDoc</strong>:</p><div class="pcenter" xmlns="http://www.w3.org/1999/xhtml"><table class="GAPDocTable"><caption class="GAPDocTable"><b>Table: </b>Predefined Entities in the <strong class="pkg">GAPDoc</strong> system</caption><tbody><tr><td colspan="1" rowspan="1" class="tdleft"><code class="code">&GAP;</code></td><td colspan="1" rowspan="1" class="tdleft"><strong class="pkg">GAP</strong></td></tr><tr><td colspan="1" rowspan="1" class="tdleft"><code class="code">&GAPDoc;</code></td><td colspan="1" rowspan="1" class="tdleft"><strong class="pkg">GAPDoc</strong></td></tr><tr><td colspan="1" rowspan="1" class="tdleft"><code class="code">&TeX;</code></td><td colspan="1" rowspan="1" class="tdleft">TeX</td></tr><tr><td colspan="1" rowspan="1" class="tdleft"><code class="code">&LaTeX;</code></td><td colspan="1" rowspan="1" class="tdleft">LaTeX</td></tr><tr><td colspan="1" rowspan="1" class="tdleft"><code class="code">&BibTeX;</code></td><td colspan="1" rowspan="1" class="tdleft">BibTeX</td></tr><tr><td colspan="1" rowspan="1" class="tdleft"><code class="code">&MeatAxe;</code></td><td colspan="1" rowspan="1" class="tdleft"><strong class="pkg">MeatAxe</strong></td></tr><tr><td colspan="1" rowspan="1" class="tdleft"><code class="code">&XGAP;</code></td><td colspan="1" rowspan="1" class="tdleft"><strong class="pkg">XGAP</strong></td></tr><tr><td colspan="1" rowspan="1" class="tdleft"><code class="code">&copyright;</code></td><td colspan="1" rowspan="1" class="tdleft">©</td></tr></tbody></table><br/><p> </p><br/>
+</div><p xmlns="http://www.w3.org/1999/xhtml">And some more for mathematical symbols: â„‚, ℤ, â„•, â„™, â„š, â„, â„.</p><div class="chlinkprevnextbot" xmlns="http://www.w3.org/1999/xhtml"> <a shape="rect" href="chap0_mj.html">[Top of Book]</a> <a shape="rect" href="chap0_mj.html#contents">[Contents]</a>  <a shape="rect" href="chap1_mj.html">[Previous Chapter]</a>  <a shape="rect" href="chapA_mj.html">[Next Chapter]</a> </div><div class="chlinkbot" xmlns="http://www.w3.org/1999/xhtml"><span class="chlink1">Goto Chapter: </span><a shape="rect" href="chap0_mj.html">Top</a> <a shape="rect" href="chap1_mj.html">1</a> <a shape="rect" href="chap2_mj.html">2</a> <a shape="rect" href="chapA_mj.html">A</a> <a shape="rect" href="chapB_mj.html">B</a> <a shape="rect" href="chapBib_mj.html">Bib</a> <a shape="rect" href="chapInd_mj.html">Ind</a> </div><hr xmlns="http://www.w3.org/1999/xhtml"/><p class="foot" xmlns="http://www.w3.org/1999/xhtml">generated by <a shape="rect" href="http://www.math.rwth-aachen.de/~Frank.Luebeck/GAPDoc">GAPDoc2HTML</a></p> </body>
</html>
\ No newline at end of file
diff --git a/export/mwscrawler/10.html b/export/mwscrawler/10.html
new file mode 100644
index 0000000..b1824ba
--- /dev/null
+++ b/export/mwscrawler/10.html
@@ -0,0 +1,18 @@
+<html>
+ <head>
+ <title>60.4-2 ANFAutomorphism</title>
+ <meta name="url" content="http://www.gap-system.org/Manuals/doc/ref/TODO#X8643D4B47A827D9D"></meta>
+ </head>
+ <body> <h5 xmlns="http://www.w3.org/1999/xhtml">60.4-2 ANFAutomorphism</h5><div class="func" xmlns="http://www.w3.org/1999/xhtml"><table width="100%" class="func"><tbody><tr><td colspan="1" rowspan="1" class="tdleft"><code class="func">‣ ANFAutomorphism</code>( <var class="Arg">F</var>, <var class="Arg">k</var> )</td><td colspan="1" rowspan="1" class="tdright">( function )</td></tr></tbody></table></div><p xmlns="http://www.w3.org/1999/xhtml">Let <var class="Arg">F</var> be an abelian number field and <var class="Arg">k</var> be an integer that is coprime to the conductor (see <code class="func">Conductor</code> (<a shape="rect" href="chap18_mj.html#X815D6EC57CBA9827"><span class="RefLink">18.1-7</span></a>)) of <var class="Arg">F</var>. Then <code class="func">ANFAutomorphism</code> returns the automorphism of <var class="Arg">F</var> that is defined as the linear extension of the map that raises each root of unity in <var class="Arg">F</var> to its <var class="Arg">k</var>-th power.</p><div class="example" xmlns="http://www.w3.org/1999/xhtml"><pre xml:space="preserve">
+<span class="GAPprompt">gap></span> <span class="GAPinput">f:= CF(25);</span>
+CF(25)
+<span class="GAPprompt">gap></span> <span class="GAPinput">alpha:= ANFAutomorphism( f, 2 );</span>
+ANFAutomorphism( CF(25), 2 )
+<span class="GAPprompt">gap></span> <span class="GAPinput">alpha^2;</span>
+ANFAutomorphism( CF(25), 4 )
+<span class="GAPprompt">gap></span> <span class="GAPinput">Order( alpha );</span>
+20
+<span class="GAPprompt">gap></span> <span class="GAPinput">E(5)^alpha;</span>
+E(5)^2
+</pre></div> </body>
+ </html>
\ No newline at end of file
diff --git a/export/mwscrawler/11.html b/export/mwscrawler/11.html
new file mode 100644
index 0000000..00c158c
--- /dev/null
+++ b/export/mwscrawler/11.html
@@ -0,0 +1,532 @@
+<html>
+ <head>
+ <title>60.4-1 GaloisGroup</title>
+ <meta name="url" content="http://www.gap-system.org/Manuals/doc/ref/TODO#X7B55A90582E818F3"></meta>
+ </head>
+ <body> <h5 xmlns="http://www.w3.org/1999/xhtml">60.4-1 GaloisGroup</h5><div class="func" xmlns="http://www.w3.org/1999/xhtml"><table width="100%" class="func"><tbody><tr><td colspan="1" rowspan="1" class="tdleft"><code class="func">‣ GaloisGroup</code>( <var class="Arg">F</var> )</td><td colspan="1" rowspan="1" class="tdright">( method )</td></tr></tbody></table></div><p xmlns="http://www.w3.org/1999/xhtml">The Galois group <math id="-3438466668951179699" display="inline" alttext="Gal(?_{n},?)" class="ltx_Math" xmlns="http://www.w3.org/1998/Math/MathML">
+ <semantics id="p1.1.m1.1a">
+ <mrow xref="p1.1.m1.1.10.cmml" id="p1.1.m1.1.10">
+ <mi xref="p1.1.m1.1.1.cmml" id="p1.1.m1.1.1">G</mi>
+ <mo xref="p1.1.m1.1.10.1.cmml" id="p1.1.m1.1.10.1">â¢</mo>
+ <mi xref="p1.1.m1.1.2.cmml" id="p1.1.m1.1.2">a</mi>
+ <mo xref="p1.1.m1.1.10.1.cmml" id="p1.1.m1.1.10.1a">â¢</mo>
+ <mi xref="p1.1.m1.1.3.cmml" id="p1.1.m1.1.3">l</mi>
+ <mo xref="p1.1.m1.1.10.1.cmml" id="p1.1.m1.1.10.1b">â¢</mo>
+ <mrow xref="p1.1.m1.1.10.2.1.cmml" id="p1.1.m1.1.10.2">
+ <mo id="p1.1.m1.1.4" stretchy="false">(</mo>
+ <msub xref="p1.1.m1.1.10.2.1.cmml" id="p1.1.m1.1.10.2.2">
+ <mi xref="p1.1.m1.1.5.cmml" id="p1.1.m1.1.5" mathvariant="normal">?</mi>
+ <mi xref="p1.1.m1.1.6.1.cmml" id="p1.1.m1.1.6.1">n</mi>
+ </msub>
+ <mo id="p1.1.m1.1.7">,</mo>
+ <mi xref="p1.1.m1.1.8.cmml" id="p1.1.m1.1.8" mathvariant="normal">?</mi>
+ <mo id="p1.1.m1.1.9" stretchy="false">)</mo>
+ </mrow>
+ </mrow>
+ <annotation-xml id="p1.1.m1.1b" encoding="MathML-Content">
+ <apply xref="p1.1.m1.1.10" id="p1.1.m1.1.10.cmml">
+ <times xref="p1.1.m1.1.10.1" id="p1.1.m1.1.10.1.cmml"/>
+ <ci xref="p1.1.m1.1.1" id="p1.1.m1.1.1.cmml">ðº</ci>
+ <ci xref="p1.1.m1.1.2" id="p1.1.m1.1.2.cmml">ð‘Ž</ci>
+ <ci xref="p1.1.m1.1.3" id="p1.1.m1.1.3.cmml">ð‘™</ci>
+ <interval xref="p1.1.m1.1.10.2" id="p1.1.m1.1.10.2.1.cmml" closure="open">
+ <apply xref="p1.1.m1.1.10.2" id="p1.1.m1.1.10.2.2.cmml">
+ <csymbol id="p1.1.m1.1.10.2.2.1.cmml" cd="ambiguous">subscript</csymbol>
+ <ci xref="p1.1.m1.1.5" id="p1.1.m1.1.5.cmml">?</ci>
+ <ci xref="p1.1.m1.1.6.1" id="p1.1.m1.1.6.1.cmml">ð‘›</ci>
+ </apply>
+ <ci xref="p1.1.m1.1.8" id="p1.1.m1.1.8.cmml">?</ci>
+ </interval>
+ </apply>
+ </annotation-xml>
+ <annotation id="p1.1.m1.1c" encoding="application/x-tex">Gal(?_{n},?)</annotation>
+ </semantics>
+</math> of the field extension <math id="-26079208874736148" display="inline" alttext="?_{n}/?" class="ltx_Math" xmlns="http://www.w3.org/1998/Math/MathML">
+ <semantics id="p1.1.m1.1a">
+ <mrow xref="p1.1.m1.1.5.cmml" id="p1.1.m1.1.5">
+ <msub xref="p1.1.m1.1.5.1.cmml" id="p1.1.m1.1.5.1">
+ <mi xref="p1.1.m1.1.1.cmml" id="p1.1.m1.1.1" mathvariant="normal">?</mi>
+ <mi xref="p1.1.m1.1.2.1.cmml" id="p1.1.m1.1.2.1">n</mi>
+ </msub>
+ <mo xref="p1.1.m1.1.3.cmml" id="p1.1.m1.1.3">/</mo>
+ <mi xref="p1.1.m1.1.4.cmml" id="p1.1.m1.1.4" mathvariant="normal">?</mi>
+ </mrow>
+ <annotation-xml id="p1.1.m1.1b" encoding="MathML-Content">
+ <apply xref="p1.1.m1.1.5" id="p1.1.m1.1.5.cmml">
+ <divide xref="p1.1.m1.1.3" id="p1.1.m1.1.3.cmml"/>
+ <apply xref="p1.1.m1.1.5.1" id="p1.1.m1.1.5.1.cmml">
+ <csymbol id="p1.1.m1.1.5.1.1.cmml" cd="ambiguous">subscript</csymbol>
+ <ci xref="p1.1.m1.1.1" id="p1.1.m1.1.1.cmml">?</ci>
+ <ci xref="p1.1.m1.1.2.1" id="p1.1.m1.1.2.1.cmml">ð‘›</ci>
+ </apply>
+ <ci xref="p1.1.m1.1.4" id="p1.1.m1.1.4.cmml">?</ci>
+ </apply>
+ </annotation-xml>
+ <annotation id="p1.1.m1.1c" encoding="application/x-tex">?_{n}/?</annotation>
+ </semantics>
+</math> is isomorphic to the group <math id="-6902481410061116804" display="inline" alttext="(?/n?)^{*}" class="ltx_Math" xmlns="http://www.w3.org/1998/Math/MathML">
+ <semantics id="p1.1.m1.1a">
+ <msup xref="p1.1.m1.1.8.cmml" id="p1.1.m1.1.8">
+ <mrow xref="p1.1.m1.1.8.2.2.cmml" id="p1.1.m1.1.8.2">
+ <mo id="p1.1.m1.1.1" stretchy="false">(</mo>
+ <mrow xref="p1.1.m1.1.8.2.2.cmml" id="p1.1.m1.1.8.2.2">
+ <mrow xref="p1.1.m1.1.8.2.2.cmml" id="p1.1.m1.1.8.2.2.2">
+ <mi xref="p1.1.m1.1.2.cmml" id="p1.1.m1.1.2" mathvariant="normal">?</mi>
+ <mo xref="p1.1.m1.1.3.cmml" id="p1.1.m1.1.3">/</mo>
+ <mi xref="p1.1.m1.1.4.cmml" id="p1.1.m1.1.4">n</mi>
+ </mrow>
+ <mo xref="p1.1.m1.1.8.2.2.1.cmml" id="p1.1.m1.1.8.2.2.1">â¢</mo>
+ <mi xref="p1.1.m1.1.5.cmml" id="p1.1.m1.1.5" mathvariant="normal">?</mi>
+ </mrow>
+ <mo id="p1.1.m1.1.6" stretchy="false">)</mo>
+ </mrow>
+ <mo xref="p1.1.m1.1.7.1.cmml" id="p1.1.m1.1.7.1">*</mo>
+ </msup>
+ <annotation-xml id="p1.1.m1.1b" encoding="MathML-Content">
+ <apply xref="p1.1.m1.1.8" id="p1.1.m1.1.8.cmml">
+ <csymbol id="p1.1.m1.1.8.1.cmml" cd="ambiguous">superscript</csymbol>
+ <apply xref="p1.1.m1.1.8.2" id="p1.1.m1.1.8.2.2.cmml">
+ <times xref="p1.1.m1.1.8.2.2.1" id="p1.1.m1.1.8.2.2.1.cmml"/>
+ <apply xref="p1.1.m1.1.8.2" id="p1.1.m1.1.8.2.2.2.cmml">
+ <divide xref="p1.1.m1.1.3" id="p1.1.m1.1.3.cmml"/>
+ <ci xref="p1.1.m1.1.2" id="p1.1.m1.1.2.cmml">?</ci>
+ <ci xref="p1.1.m1.1.4" id="p1.1.m1.1.4.cmml">ð‘›</ci>
+ </apply>
+ <ci xref="p1.1.m1.1.5" id="p1.1.m1.1.5.cmml">?</ci>
+ </apply>
+ <times xref="p1.1.m1.1.7.1" id="p1.1.m1.1.7.1.cmml"/>
+ </apply>
+ </annotation-xml>
+ <annotation id="p1.1.m1.1c" encoding="application/x-tex">(?/n?)^{*}</annotation>
+ </semantics>
+</math> of prime residues modulo <math id="-8154649107965225231" display="inline" alttext="n" class="ltx_Math" xmlns="http://www.w3.org/1998/Math/MathML">
+ <semantics id="p1.1.m1.1a">
+ <mi xref="p1.1.m1.1.1.cmml" id="p1.1.m1.1.1">n</mi>
+ <annotation-xml id="p1.1.m1.1b" encoding="MathML-Content">
+ <ci xref="p1.1.m1.1.1" id="p1.1.m1.1.1.cmml">ð‘›</ci>
+ </annotation-xml>
+ <annotation id="p1.1.m1.1c" encoding="application/x-tex">n</annotation>
+ </semantics>
+</math>, via the isomorphism <math id="1417693036080077930" display="inline" alttext="(?/n?)^{*}\rightarrow Gal(?_{n},?)" class="ltx_Math" xmlns="http://www.w3.org/1998/Math/MathML">
+ <semantics id="p1.1.m1.1a">
+ <mrow xref="p1.1.m1.1.18.cmml" id="p1.1.m1.1.18">
+ <msup xref="p1.1.m1.1.18.1.cmml" id="p1.1.m1.1.18.1">
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+ <mo id="p1.1.m1.1.15">,</mo>
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+ <mo id="p1.1.m1.1.17" stretchy="false">)</mo>
+ </mrow>
+ </mrow>
+ </mrow>
+ <annotation-xml id="p1.1.m1.1b" encoding="MathML-Content">
+ <apply xref="p1.1.m1.1.18" id="p1.1.m1.1.18.cmml">
+ <ci xref="p1.1.m1.1.8" id="p1.1.m1.1.8.cmml">→</ci>
+ <apply xref="p1.1.m1.1.18.1" id="p1.1.m1.1.18.1.cmml">
+ <csymbol id="p1.1.m1.1.18.1.1.cmml" cd="ambiguous">superscript</csymbol>
+ <apply xref="p1.1.m1.1.18.1.2" id="p1.1.m1.1.18.1.2.2.cmml">
+ <times xref="p1.1.m1.1.18.1.2.2.1" id="p1.1.m1.1.18.1.2.2.1.cmml"/>
+ <apply xref="p1.1.m1.1.18.1.2" id="p1.1.m1.1.18.1.2.2.2.cmml">
+ <divide xref="p1.1.m1.1.3" id="p1.1.m1.1.3.cmml"/>
+ <ci xref="p1.1.m1.1.2" id="p1.1.m1.1.2.cmml">?</ci>
+ <ci xref="p1.1.m1.1.4" id="p1.1.m1.1.4.cmml">ð‘›</ci>
+ </apply>
+ <ci xref="p1.1.m1.1.5" id="p1.1.m1.1.5.cmml">?</ci>
+ </apply>
+ <times xref="p1.1.m1.1.7.1" id="p1.1.m1.1.7.1.cmml"/>
+ </apply>
+ <apply xref="p1.1.m1.1.18.2" id="p1.1.m1.1.18.2.cmml">
+ <times xref="p1.1.m1.1.18.2.1" id="p1.1.m1.1.18.2.1.cmml"/>
+ <ci xref="p1.1.m1.1.9" id="p1.1.m1.1.9.cmml">ðº</ci>
+ <ci xref="p1.1.m1.1.10" id="p1.1.m1.1.10.cmml">ð‘Ž</ci>
+ <ci xref="p1.1.m1.1.11" id="p1.1.m1.1.11.cmml">ð‘™</ci>
+ <interval xref="p1.1.m1.1.18.2.2" id="p1.1.m1.1.18.2.2.1.cmml" closure="open">
+ <apply xref="p1.1.m1.1.18.2.2" id="p1.1.m1.1.18.2.2.2.cmml">
+ <csymbol id="p1.1.m1.1.18.2.2.2.1.cmml" cd="ambiguous">subscript</csymbol>
+ <ci xref="p1.1.m1.1.13" id="p1.1.m1.1.13.cmml">?</ci>
+ <ci xref="p1.1.m1.1.14.1" id="p1.1.m1.1.14.1.cmml">ð‘›</ci>
+ </apply>
+ <ci xref="p1.1.m1.1.16" id="p1.1.m1.1.16.cmml">?</ci>
+ </interval>
+ </apply>
+ </apply>
+ </annotation-xml>
+ <annotation id="p1.1.m1.1c" encoding="application/x-tex">(?/n?)^{*}\rightarrow Gal(?_{n},?)</annotation>
+ </semantics>
+</math> that is defined by <math id="-5519793287242170455" display="inline" alttext="k+n?\mapsto(z\mapsto z^{*k})" class="ltx_Math" xmlns="http://www.w3.org/1998/Math/MathML">
+ <semantics id="p1.1.m1.1a">
+ <mrow id="p1.1.m1.1b">
+ <mi id="p1.1.m1.1.1">k</mi>
+ <mo xref="p1.1.m1.1.2.cmml" id="p1.1.m1.1.2">+</mo>
+ <mi id="p1.1.m1.1.3">n</mi>
+ <mi id="p1.1.m1.1.4" mathvariant="normal">?</mi>
+ <mo xref="p1.1.m1.1.5.cmml" id="p1.1.m1.1.5">↦</mo>
+ <mrow xref="p1.1.m1.1.10.cmml" id="p1.1.m1.1.10">
+ <mo xref="p1.1.m1.1.6.cmml" id="p1.1.m1.1.6" stretchy="false">(</mo>
+ <mi id="p1.1.m1.1.7">z</mi>
+ <mo xref="p1.1.m1.1.8.cmml" id="p1.1.m1.1.8">↦</mo>
+ <msup xref="p1.1.m1.1.12.cmml" id="p1.1.m1.1.12">
+ <mi xref="p1.1.m1.1.9.cmml" id="p1.1.m1.1.9">z</mi>
+ <mrow xref="p1.1.m1.1.10.1.cmml" id="p1.1.m1.1.10.1">
+ <mi xref="p1.1.m1.1.10.1.3.cmml" id="p1.1.m1.1.10.1.3"/>
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+ <mi xref="p1.1.m1.1.10.1.2.cmml" id="p1.1.m1.1.10.1.2">k</mi>
+ </mrow>
+ </msup>
+ <mo xref="p1.1.m1.1.11.cmml" id="p1.1.m1.1.11" stretchy="false">)</mo>
+ </mrow>
+ </mrow>
+ <annotation-xml id="p1.1.m1.1c" encoding="MathML-Content">
+ <cerror id="p1.1.m1.1d">
+ <csymbol id="p1.1.m1.1e" cd="ambiguous">fragments</csymbol>
+ <csymbol id="p1.1.m1.1f" cd="unknown">k</csymbol>
+ <plus xref="p1.1.m1.1.2" id="p1.1.m1.1.2.cmml"/>
+ <csymbol id="p1.1.m1.1g" cd="unknown">n</csymbol>
+ <csymbol id="p1.1.m1.1h" cd="unknown">?</csymbol>
+ <csymbol xref="p1.1.m1.1.5" id="p1.1.m1.1.5.cmml" cd="latexml">maps-to</csymbol>
+ <cerror xref="p1.1.m1.1.10" id="p1.1.m1.1.10.cmml">
+ <csymbol xref="p1.1.m1.1.10" id="p1.1.m1.1.10a.cmml" cd="ambiguous">fragments</csymbol>
+ <ci xref="p1.1.m1.1.6" id="p1.1.m1.1.6.cmml">(</ci>
+ <csymbol xref="p1.1.m1.1.10" id="p1.1.m1.1.10b.cmml" cd="unknown">z</csymbol>
+ <csymbol xref="p1.1.m1.1.8" id="p1.1.m1.1.8.cmml" cd="latexml">maps-to</csymbol>
+ <apply xref="p1.1.m1.1.12" id="p1.1.m1.1.12.cmml">
+ <csymbol id="p1.1.m1.1.12.1.cmml" cd="ambiguous">superscript</csymbol>
+ <ci xref="p1.1.m1.1.9" id="p1.1.m1.1.9.cmml">ð‘§</ci>
+ <apply xref="p1.1.m1.1.10.1" id="p1.1.m1.1.10.1.cmml">
+ <times xref="p1.1.m1.1.10.1.1" id="p1.1.m1.1.10.1.1.cmml"/>
+ <csymbol xref="p1.1.m1.1.10.1.3" id="p1.1.m1.1.10.1.3.cmml" cd="latexml">absent</csymbol>
+ <ci xref="p1.1.m1.1.10.1.2" id="p1.1.m1.1.10.1.2.cmml">ð‘˜</ci>
+ </apply>
+ </apply>
+ <ci xref="p1.1.m1.1.11" id="p1.1.m1.1.11.cmml">)</ci>
+ </cerror>
+ </cerror>
+ </annotation-xml>
+ <annotation id="p1.1.m1.1i" encoding="application/x-tex">k+n?\mapsto(z\mapsto z^{*k})</annotation>
+ </semantics>
+</math>.</p><p xmlns="http://www.w3.org/1999/xhtml">The Galois group of the field extension <math id="405475826120630514" display="inline" alttext="?_{n}/L" class="ltx_Math" xmlns="http://www.w3.org/1998/Math/MathML">
+ <semantics id="p1.1.m1.1a">
+ <mrow xref="p1.1.m1.1.5.cmml" id="p1.1.m1.1.5">
+ <msub xref="p1.1.m1.1.5.1.cmml" id="p1.1.m1.1.5.1">
+ <mi xref="p1.1.m1.1.1.cmml" id="p1.1.m1.1.1" mathvariant="normal">?</mi>
+ <mi xref="p1.1.m1.1.2.1.cmml" id="p1.1.m1.1.2.1">n</mi>
+ </msub>
+ <mo xref="p1.1.m1.1.3.cmml" id="p1.1.m1.1.3">/</mo>
+ <mi xref="p1.1.m1.1.4.cmml" id="p1.1.m1.1.4">L</mi>
+ </mrow>
+ <annotation-xml id="p1.1.m1.1b" encoding="MathML-Content">
+ <apply xref="p1.1.m1.1.5" id="p1.1.m1.1.5.cmml">
+ <divide xref="p1.1.m1.1.3" id="p1.1.m1.1.3.cmml"/>
+ <apply xref="p1.1.m1.1.5.1" id="p1.1.m1.1.5.1.cmml">
+ <csymbol id="p1.1.m1.1.5.1.1.cmml" cd="ambiguous">subscript</csymbol>
+ <ci xref="p1.1.m1.1.1" id="p1.1.m1.1.1.cmml">?</ci>
+ <ci xref="p1.1.m1.1.2.1" id="p1.1.m1.1.2.1.cmml">ð‘›</ci>
+ </apply>
+ <ci xref="p1.1.m1.1.4" id="p1.1.m1.1.4.cmml">ð¿</ci>
+ </apply>
+ </annotation-xml>
+ <annotation id="p1.1.m1.1c" encoding="application/x-tex">?_{n}/L</annotation>
+ </semantics>
+</math> with any abelian number field <math id="-670034400719986221" display="inline" alttext="L\subseteq?_{n}" class="ltx_Math" xmlns="http://www.w3.org/1998/Math/MathML">
+ <semantics id="p1.1.m1.1a">
+ <mrow xref="p1.1.m1.1.5.cmml" id="p1.1.m1.1.5">
+ <mi xref="p1.1.m1.1.1.cmml" id="p1.1.m1.1.1">L</mi>
+ <mo xref="p1.1.m1.1.2.cmml" id="p1.1.m1.1.2">⊆</mo>
+ <msub xref="p1.1.m1.1.5.1.cmml" id="p1.1.m1.1.5.1">
+ <mi xref="p1.1.m1.1.3.cmml" id="p1.1.m1.1.3" mathvariant="normal">?</mi>
+ <mi xref="p1.1.m1.1.4.1.cmml" id="p1.1.m1.1.4.1">n</mi>
+ </msub>
+ </mrow>
+ <annotation-xml id="p1.1.m1.1b" encoding="MathML-Content">
+ <apply xref="p1.1.m1.1.5" id="p1.1.m1.1.5.cmml">
+ <subset xref="p1.1.m1.1.2" id="p1.1.m1.1.2.cmml"/>
+ <ci xref="p1.1.m1.1.1" id="p1.1.m1.1.1.cmml">ð¿</ci>
+ <apply xref="p1.1.m1.1.5.1" id="p1.1.m1.1.5.1.cmml">
+ <csymbol id="p1.1.m1.1.5.1.1.cmml" cd="ambiguous">subscript</csymbol>
+ <ci xref="p1.1.m1.1.3" id="p1.1.m1.1.3.cmml">?</ci>
+ <ci xref="p1.1.m1.1.4.1" id="p1.1.m1.1.4.1.cmml">ð‘›</ci>
+ </apply>
+ </apply>
+ </annotation-xml>
+ <annotation id="p1.1.m1.1c" encoding="application/x-tex">L\subseteq?_{n}</annotation>
+ </semantics>
+</math> is simply the factor group of <math id="-7456608632428795393" display="inline" alttext="Gal(?_{n},?)" class="ltx_Math" xmlns="http://www.w3.org/1998/Math/MathML">
+ <semantics id="p1.1.m1.1a">
+ <mrow xref="p1.1.m1.1.10.cmml" id="p1.1.m1.1.10">
+ <mi xref="p1.1.m1.1.1.cmml" id="p1.1.m1.1.1">G</mi>
+ <mo xref="p1.1.m1.1.10.1.cmml" id="p1.1.m1.1.10.1">â¢</mo>
+ <mi xref="p1.1.m1.1.2.cmml" id="p1.1.m1.1.2">a</mi>
+ <mo xref="p1.1.m1.1.10.1.cmml" id="p1.1.m1.1.10.1a">â¢</mo>
+ <mi xref="p1.1.m1.1.3.cmml" id="p1.1.m1.1.3">l</mi>
+ <mo xref="p1.1.m1.1.10.1.cmml" id="p1.1.m1.1.10.1b">â¢</mo>
+ <mrow xref="p1.1.m1.1.10.2.1.cmml" id="p1.1.m1.1.10.2">
+ <mo id="p1.1.m1.1.4" stretchy="false">(</mo>
+ <msub xref="p1.1.m1.1.10.2.1.cmml" id="p1.1.m1.1.10.2.2">
+ <mi xref="p1.1.m1.1.5.cmml" id="p1.1.m1.1.5" mathvariant="normal">?</mi>
+ <mi xref="p1.1.m1.1.6.1.cmml" id="p1.1.m1.1.6.1">n</mi>
+ </msub>
+ <mo id="p1.1.m1.1.7">,</mo>
+ <mi xref="p1.1.m1.1.8.cmml" id="p1.1.m1.1.8" mathvariant="normal">?</mi>
+ <mo id="p1.1.m1.1.9" stretchy="false">)</mo>
+ </mrow>
+ </mrow>
+ <annotation-xml id="p1.1.m1.1b" encoding="MathML-Content">
+ <apply xref="p1.1.m1.1.10" id="p1.1.m1.1.10.cmml">
+ <times xref="p1.1.m1.1.10.1" id="p1.1.m1.1.10.1.cmml"/>
+ <ci xref="p1.1.m1.1.1" id="p1.1.m1.1.1.cmml">ðº</ci>
+ <ci xref="p1.1.m1.1.2" id="p1.1.m1.1.2.cmml">ð‘Ž</ci>
+ <ci xref="p1.1.m1.1.3" id="p1.1.m1.1.3.cmml">ð‘™</ci>
+ <interval xref="p1.1.m1.1.10.2" id="p1.1.m1.1.10.2.1.cmml" closure="open">
+ <apply xref="p1.1.m1.1.10.2" id="p1.1.m1.1.10.2.2.cmml">
+ <csymbol id="p1.1.m1.1.10.2.2.1.cmml" cd="ambiguous">subscript</csymbol>
+ <ci xref="p1.1.m1.1.5" id="p1.1.m1.1.5.cmml">?</ci>
+ <ci xref="p1.1.m1.1.6.1" id="p1.1.m1.1.6.1.cmml">ð‘›</ci>
+ </apply>
+ <ci xref="p1.1.m1.1.8" id="p1.1.m1.1.8.cmml">?</ci>
+ </interval>
+ </apply>
+ </annotation-xml>
+ <annotation id="p1.1.m1.1c" encoding="application/x-tex">Gal(?_{n},?)</annotation>
+ </semantics>
+</math> modulo the stabilizer of <math id="-71664830046701079" display="inline" alttext="L" class="ltx_Math" xmlns="http://www.w3.org/1998/Math/MathML">
+ <semantics id="p1.1.m1.1a">
+ <mi xref="p1.1.m1.1.1.cmml" id="p1.1.m1.1.1">L</mi>
+ <annotation-xml id="p1.1.m1.1b" encoding="MathML-Content">
+ <ci xref="p1.1.m1.1.1" id="p1.1.m1.1.1.cmml">ð¿</ci>
+ </annotation-xml>
+ <annotation id="p1.1.m1.1c" encoding="application/x-tex">L</annotation>
+ </semantics>
+</math>, and the Galois group of <math id="5031197143387100812" display="inline" alttext="L/L^{\prime}" class="ltx_Math" xmlns="http://www.w3.org/1998/Math/MathML">
+ <semantics id="p1.1.m1.1a">
+ <mrow xref="p1.1.m1.1.5.cmml" id="p1.1.m1.1.5">
+ <mi xref="p1.1.m1.1.1.cmml" id="p1.1.m1.1.1">L</mi>
+ <mo xref="p1.1.m1.1.2.cmml" id="p1.1.m1.1.2">/</mo>
+ <msup xref="p1.1.m1.1.5.1.cmml" id="p1.1.m1.1.5.1">
+ <mi xref="p1.1.m1.1.3.cmml" id="p1.1.m1.1.3">L</mi>
+ <mo xref="p1.1.m1.1.4.1.cmml" id="p1.1.m1.1.4.1">′</mo>
+ </msup>
+ </mrow>
+ <annotation-xml id="p1.1.m1.1b" encoding="MathML-Content">
+ <apply xref="p1.1.m1.1.5" id="p1.1.m1.1.5.cmml">
+ <divide xref="p1.1.m1.1.2" id="p1.1.m1.1.2.cmml"/>
+ <ci xref="p1.1.m1.1.1" id="p1.1.m1.1.1.cmml">ð¿</ci>
+ <apply xref="p1.1.m1.1.5.1" id="p1.1.m1.1.5.1.cmml">
+ <csymbol id="p1.1.m1.1.5.1.1.cmml" cd="ambiguous">superscript</csymbol>
+ <ci xref="p1.1.m1.1.3" id="p1.1.m1.1.3.cmml">ð¿</ci>
+ <ci xref="p1.1.m1.1.4.1" id="p1.1.m1.1.4.1.cmml">′</ci>
+ </apply>
+ </apply>
+ </annotation-xml>
+ <annotation id="p1.1.m1.1c" encoding="application/x-tex">L/L^{\prime}</annotation>
+ </semantics>
+</math>, with <math id="3382457761506254890" display="inline" alttext="L^{\prime}" class="ltx_Math" xmlns="http://www.w3.org/1998/Math/MathML">
+ <semantics id="p1.1.m1.1a">
+ <msup xref="p1.1.m1.1.3.cmml" id="p1.1.m1.1.3">
+ <mi xref="p1.1.m1.1.1.cmml" id="p1.1.m1.1.1">L</mi>
+ <mo xref="p1.1.m1.1.2.1.cmml" id="p1.1.m1.1.2.1">′</mo>
+ </msup>
+ <annotation-xml id="p1.1.m1.1b" encoding="MathML-Content">
+ <apply xref="p1.1.m1.1.3" id="p1.1.m1.1.3.cmml">
+ <csymbol id="p1.1.m1.1.3.1.cmml" cd="ambiguous">superscript</csymbol>
+ <ci xref="p1.1.m1.1.1" id="p1.1.m1.1.1.cmml">ð¿</ci>
+ <ci xref="p1.1.m1.1.2.1" id="p1.1.m1.1.2.1.cmml">′</ci>
+ </apply>
+ </annotation-xml>
+ <annotation id="p1.1.m1.1c" encoding="application/x-tex">L^{\prime}</annotation>
+ </semantics>
+</math> an abelian number field contained in <math id="-6935840984003647815" display="inline" alttext="L" class="ltx_Math" xmlns="http://www.w3.org/1998/Math/MathML">
+ <semantics id="p1.1.m1.1a">
+ <mi xref="p1.1.m1.1.1.cmml" id="p1.1.m1.1.1">L</mi>
+ <annotation-xml id="p1.1.m1.1b" encoding="MathML-Content">
+ <ci xref="p1.1.m1.1.1" id="p1.1.m1.1.1.cmml">ð¿</ci>
+ </annotation-xml>
+ <annotation id="p1.1.m1.1c" encoding="application/x-tex">L</annotation>
+ </semantics>
+</math>, is the subgroup in this group that stabilizes <math id="-1110171846552566727" display="inline" alttext="L^{\prime}" class="ltx_Math" xmlns="http://www.w3.org/1998/Math/MathML">
+ <semantics id="p1.1.m1.1a">
+ <msup xref="p1.1.m1.1.3.cmml" id="p1.1.m1.1.3">
+ <mi xref="p1.1.m1.1.1.cmml" id="p1.1.m1.1.1">L</mi>
+ <mo xref="p1.1.m1.1.2.1.cmml" id="p1.1.m1.1.2.1">′</mo>
+ </msup>
+ <annotation-xml id="p1.1.m1.1b" encoding="MathML-Content">
+ <apply xref="p1.1.m1.1.3" id="p1.1.m1.1.3.cmml">
+ <csymbol id="p1.1.m1.1.3.1.cmml" cd="ambiguous">superscript</csymbol>
+ <ci xref="p1.1.m1.1.1" id="p1.1.m1.1.1.cmml">ð¿</ci>
+ <ci xref="p1.1.m1.1.2.1" id="p1.1.m1.1.2.1.cmml">′</ci>
+ </apply>
+ </annotation-xml>
+ <annotation id="p1.1.m1.1c" encoding="application/x-tex">L^{\prime}</annotation>
+ </semantics>
+</math>. These groups are easily described in terms of <math id="-6776385823676840604" display="inline" alttext="(?/n?)^{*}" class="ltx_Math" xmlns="http://www.w3.org/1998/Math/MathML">
+ <semantics id="p1.1.m1.1a">
+ <msup xref="p1.1.m1.1.8.cmml" id="p1.1.m1.1.8">
+ <mrow xref="p1.1.m1.1.8.2.2.cmml" id="p1.1.m1.1.8.2">
+ <mo id="p1.1.m1.1.1" stretchy="false">(</mo>
+ <mrow xref="p1.1.m1.1.8.2.2.cmml" id="p1.1.m1.1.8.2.2">
+ <mrow xref="p1.1.m1.1.8.2.2.cmml" id="p1.1.m1.1.8.2.2.2">
+ <mi xref="p1.1.m1.1.2.cmml" id="p1.1.m1.1.2" mathvariant="normal">?</mi>
+ <mo xref="p1.1.m1.1.3.cmml" id="p1.1.m1.1.3">/</mo>
+ <mi xref="p1.1.m1.1.4.cmml" id="p1.1.m1.1.4">n</mi>
+ </mrow>
+ <mo xref="p1.1.m1.1.8.2.2.1.cmml" id="p1.1.m1.1.8.2.2.1">â¢</mo>
+ <mi xref="p1.1.m1.1.5.cmml" id="p1.1.m1.1.5" mathvariant="normal">?</mi>
+ </mrow>
+ <mo id="p1.1.m1.1.6" stretchy="false">)</mo>
+ </mrow>
+ <mo xref="p1.1.m1.1.7.1.cmml" id="p1.1.m1.1.7.1">*</mo>
+ </msup>
+ <annotation-xml id="p1.1.m1.1b" encoding="MathML-Content">
+ <apply xref="p1.1.m1.1.8" id="p1.1.m1.1.8.cmml">
+ <csymbol id="p1.1.m1.1.8.1.cmml" cd="ambiguous">superscript</csymbol>
+ <apply xref="p1.1.m1.1.8.2" id="p1.1.m1.1.8.2.2.cmml">
+ <times xref="p1.1.m1.1.8.2.2.1" id="p1.1.m1.1.8.2.2.1.cmml"/>
+ <apply xref="p1.1.m1.1.8.2" id="p1.1.m1.1.8.2.2.2.cmml">
+ <divide xref="p1.1.m1.1.3" id="p1.1.m1.1.3.cmml"/>
+ <ci xref="p1.1.m1.1.2" id="p1.1.m1.1.2.cmml">?</ci>
+ <ci xref="p1.1.m1.1.4" id="p1.1.m1.1.4.cmml">ð‘›</ci>
+ </apply>
+ <ci xref="p1.1.m1.1.5" id="p1.1.m1.1.5.cmml">?</ci>
+ </apply>
+ <times xref="p1.1.m1.1.7.1" id="p1.1.m1.1.7.1.cmml"/>
+ </apply>
+ </annotation-xml>
+ <annotation id="p1.1.m1.1c" encoding="application/x-tex">(?/n?)^{*}</annotation>
+ </semantics>
+</math>. Generators of <math id="8398021252958325998" display="inline" alttext="(?/n?)^{*}" class="ltx_Math" xmlns="http://www.w3.org/1998/Math/MathML">
+ <semantics id="p1.1.m1.1a">
+ <msup xref="p1.1.m1.1.8.cmml" id="p1.1.m1.1.8">
+ <mrow xref="p1.1.m1.1.8.2.2.cmml" id="p1.1.m1.1.8.2">
+ <mo id="p1.1.m1.1.1" stretchy="false">(</mo>
+ <mrow xref="p1.1.m1.1.8.2.2.cmml" id="p1.1.m1.1.8.2.2">
+ <mrow xref="p1.1.m1.1.8.2.2.cmml" id="p1.1.m1.1.8.2.2.2">
+ <mi xref="p1.1.m1.1.2.cmml" id="p1.1.m1.1.2" mathvariant="normal">?</mi>
+ <mo xref="p1.1.m1.1.3.cmml" id="p1.1.m1.1.3">/</mo>
+ <mi xref="p1.1.m1.1.4.cmml" id="p1.1.m1.1.4">n</mi>
+ </mrow>
+ <mo xref="p1.1.m1.1.8.2.2.1.cmml" id="p1.1.m1.1.8.2.2.1">â¢</mo>
+ <mi xref="p1.1.m1.1.5.cmml" id="p1.1.m1.1.5" mathvariant="normal">?</mi>
+ </mrow>
+ <mo id="p1.1.m1.1.6" stretchy="false">)</mo>
+ </mrow>
+ <mo xref="p1.1.m1.1.7.1.cmml" id="p1.1.m1.1.7.1">*</mo>
+ </msup>
+ <annotation-xml id="p1.1.m1.1b" encoding="MathML-Content">
+ <apply xref="p1.1.m1.1.8" id="p1.1.m1.1.8.cmml">
+ <csymbol id="p1.1.m1.1.8.1.cmml" cd="ambiguous">superscript</csymbol>
+ <apply xref="p1.1.m1.1.8.2" id="p1.1.m1.1.8.2.2.cmml">
+ <times xref="p1.1.m1.1.8.2.2.1" id="p1.1.m1.1.8.2.2.1.cmml"/>
+ <apply xref="p1.1.m1.1.8.2" id="p1.1.m1.1.8.2.2.2.cmml">
+ <divide xref="p1.1.m1.1.3" id="p1.1.m1.1.3.cmml"/>
+ <ci xref="p1.1.m1.1.2" id="p1.1.m1.1.2.cmml">?</ci>
+ <ci xref="p1.1.m1.1.4" id="p1.1.m1.1.4.cmml">ð‘›</ci>
+ </apply>
+ <ci xref="p1.1.m1.1.5" id="p1.1.m1.1.5.cmml">?</ci>
+ </apply>
+ <times xref="p1.1.m1.1.7.1" id="p1.1.m1.1.7.1.cmml"/>
+ </apply>
+ </annotation-xml>
+ <annotation id="p1.1.m1.1c" encoding="application/x-tex">(?/n?)^{*}</annotation>
+ </semantics>
+</math> can be computed using <code class="func">GeneratorsPrimeResidues</code> (<a shape="rect" href="chap15_mj.html#X7D191CF67E5018BE"><span class="RefLink">15.2-4</span></a>).</p><p xmlns="http://www.w3.org/1999/xhtml">In <strong class="pkg">GAP</strong>, a field extension <math id="1054280403245167268" display="inline" alttext="L/L^{\prime}" class="ltx_Math" xmlns="http://www.w3.org/1998/Math/MathML">
+ <semantics id="p1.1.m1.1a">
+ <mrow xref="p1.1.m1.1.5.cmml" id="p1.1.m1.1.5">
+ <mi xref="p1.1.m1.1.1.cmml" id="p1.1.m1.1.1">L</mi>
+ <mo xref="p1.1.m1.1.2.cmml" id="p1.1.m1.1.2">/</mo>
+ <msup xref="p1.1.m1.1.5.1.cmml" id="p1.1.m1.1.5.1">
+ <mi xref="p1.1.m1.1.3.cmml" id="p1.1.m1.1.3">L</mi>
+ <mo xref="p1.1.m1.1.4.1.cmml" id="p1.1.m1.1.4.1">′</mo>
+ </msup>
+ </mrow>
+ <annotation-xml id="p1.1.m1.1b" encoding="MathML-Content">
+ <apply xref="p1.1.m1.1.5" id="p1.1.m1.1.5.cmml">
+ <divide xref="p1.1.m1.1.2" id="p1.1.m1.1.2.cmml"/>
+ <ci xref="p1.1.m1.1.1" id="p1.1.m1.1.1.cmml">ð¿</ci>
+ <apply xref="p1.1.m1.1.5.1" id="p1.1.m1.1.5.1.cmml">
+ <csymbol id="p1.1.m1.1.5.1.1.cmml" cd="ambiguous">superscript</csymbol>
+ <ci xref="p1.1.m1.1.3" id="p1.1.m1.1.3.cmml">ð¿</ci>
+ <ci xref="p1.1.m1.1.4.1" id="p1.1.m1.1.4.1.cmml">′</ci>
+ </apply>
+ </apply>
+ </annotation-xml>
+ <annotation id="p1.1.m1.1c" encoding="application/x-tex">L/L^{\prime}</annotation>
+ </semantics>
+</math> is given by the field object <math id="-841083629733370542" display="inline" alttext="L" class="ltx_Math" xmlns="http://www.w3.org/1998/Math/MathML">
+ <semantics id="p1.1.m1.1a">
+ <mi xref="p1.1.m1.1.1.cmml" id="p1.1.m1.1.1">L</mi>
+ <annotation-xml id="p1.1.m1.1b" encoding="MathML-Content">
+ <ci xref="p1.1.m1.1.1" id="p1.1.m1.1.1.cmml">ð¿</ci>
+ </annotation-xml>
+ <annotation id="p1.1.m1.1c" encoding="application/x-tex">L</annotation>
+ </semantics>
+</math> with <code class="func">LeftActingDomain</code> (<a shape="rect" href="chap57_mj.html#X86F070E0807DC34E"><span class="RefLink">57.1-11</span></a>) value <math id="6578790070330986443" display="inline" alttext="L^{\prime}" class="ltx_Math" xmlns="http://www.w3.org/1998/Math/MathML">
+ <semantics id="p1.1.m1.1a">
+ <msup xref="p1.1.m1.1.3.cmml" id="p1.1.m1.1.3">
+ <mi xref="p1.1.m1.1.1.cmml" id="p1.1.m1.1.1">L</mi>
+ <mo xref="p1.1.m1.1.2.1.cmml" id="p1.1.m1.1.2.1">′</mo>
+ </msup>
+ <annotation-xml id="p1.1.m1.1b" encoding="MathML-Content">
+ <apply xref="p1.1.m1.1.3" id="p1.1.m1.1.3.cmml">
+ <csymbol id="p1.1.m1.1.3.1.cmml" cd="ambiguous">superscript</csymbol>
+ <ci xref="p1.1.m1.1.1" id="p1.1.m1.1.1.cmml">ð¿</ci>
+ <ci xref="p1.1.m1.1.2.1" id="p1.1.m1.1.2.1.cmml">′</ci>
+ </apply>
+ </annotation-xml>
+ <annotation id="p1.1.m1.1c" encoding="application/x-tex">L^{\prime}</annotation>
+ </semantics>
+</math> (see <a shape="rect" href="chap60_mj.html#X7D2421AC8491D2BE"><span class="RefLink">60.3</span></a>).</p><div class="example" xmlns="http://www.w3.org/1999/xhtml"><pre xml:space="preserve">
+<span class="GAPprompt">gap></span> <span class="GAPinput">f:= CF(15);</span>
+CF(15)
+<span class="GAPprompt">gap></span> <span class="GAPinput">g:= GaloisGroup( f );</span>
+<group with 2 generators>
+<span class="GAPprompt">gap></span> <span class="GAPinput">Size( g ); IsCyclic( g ); IsAbelian( g );</span>
+8
+false
+true
+<span class="GAPprompt">gap></span> <span class="GAPinput">Action( g, NormalBase( f ), OnPoints );</span>
+Group([ (1,6)(2,4)(3,8)(5,7), (1,4,3,7)(2,8,5,6) ])
+</pre></div><p xmlns="http://www.w3.org/1999/xhtml">The following example shows Galois groups of a cyclotomic field and of a proper subfield that is not a cyclotomic field.</p><div class="example" xmlns="http://www.w3.org/1999/xhtml"><pre xml:space="preserve">
+<span class="GAPprompt">gap></span> <span class="GAPinput">gens1:= GeneratorsOfGroup( GaloisGroup( CF(5) ) );</span>
+[ ANFAutomorphism( CF(5), 2 ) ]
+<span class="GAPprompt">gap></span> <span class="GAPinput">gens2:= GeneratorsOfGroup( GaloisGroup( Field( Sqrt(5) ) ) );</span>
+[ ANFAutomorphism( NF(5,[ 1, 4 ]), 2 ) ]
+<span class="GAPprompt">gap></span> <span class="GAPinput">Order( gens1[1] ); Order( gens2[1] );</span>
+4
+2
+<span class="GAPprompt">gap></span> <span class="GAPinput">Sqrt(5)^gens1[1] = Sqrt(5)^gens2[1];</span>
+true
+</pre></div><p xmlns="http://www.w3.org/1999/xhtml">The following example shows the Galois group of a cyclotomic field over a non-cyclotomic field.</p><div class="example" xmlns="http://www.w3.org/1999/xhtml"><pre xml:space="preserve">
+<span class="GAPprompt">gap></span> <span class="GAPinput">g:= GaloisGroup( AsField( Field( [ Sqrt(5) ] ), CF(5) ) );</span>
+<group with 1 generators>
+<span class="GAPprompt">gap></span> <span class="GAPinput">gens:= GeneratorsOfGroup( g );</span>
+[ ANFAutomorphism( AsField( NF(5,[ 1, 4 ]), CF(5) ), 4 ) ]
+<span class="GAPprompt">gap></span> <span class="GAPinput">x:= last[1];; x^2;</span>
+IdentityMapping( AsField( NF(5,[ 1, 4 ]), CF(5) ) )
+</pre></div> </body>
+ </html>
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@@ -0,0 +1,349 @@
+<html>
+ <head>
+ <title>60.4 Galois Groups of Abelian Number Fields</title>
+ <meta name="url" content="http://www.gap-system.org/Manuals/doc/ref/TODO#X7E4AB4B17C7BA10C"></meta>
+ </head>
+ <body> <h4 xmlns="http://www.w3.org/1999/xhtml">60.4 <span class="Heading">Galois Groups of Abelian Number Fields</span></h4><p xmlns="http://www.w3.org/1999/xhtml">The field automorphisms of the cyclotomic field <math id="-8226334041509413860" display="inline" alttext="?_{n}" class="ltx_Math" xmlns="http://www.w3.org/1998/Math/MathML">
+ <semantics id="p1.1.m1.1a">
+ <msub xref="p1.1.m1.1.3.cmml" id="p1.1.m1.1.3">
+ <mi xref="p1.1.m1.1.1.cmml" id="p1.1.m1.1.1" mathvariant="normal">?</mi>
+ <mi xref="p1.1.m1.1.2.1.cmml" id="p1.1.m1.1.2.1">n</mi>
+ </msub>
+ <annotation-xml id="p1.1.m1.1b" encoding="MathML-Content">
+ <apply xref="p1.1.m1.1.3" id="p1.1.m1.1.3.cmml">
+ <csymbol id="p1.1.m1.1.3.1.cmml" cd="ambiguous">subscript</csymbol>
+ <ci xref="p1.1.m1.1.1" id="p1.1.m1.1.1.cmml">?</ci>
+ <ci xref="p1.1.m1.1.2.1" id="p1.1.m1.1.2.1.cmml">ð‘›</ci>
+ </apply>
+ </annotation-xml>
+ <annotation id="p1.1.m1.1c" encoding="application/x-tex">?_{n}</annotation>
+ </semantics>
+</math> (see Chapter <a shape="rect" href="chap18_mj.html#X7DFC03C187DE4841"><span class="RefLink">18</span></a>) are given by the linear maps <math id="-91427219792583270" display="inline" alttext="*k" class="ltx_Math" xmlns="http://www.w3.org/1998/Math/MathML">
+ <semantics id="p1.1.m1.1a">
+ <mrow xref="p1.1.m1.1.3.cmml" id="p1.1.m1.1.3">
+ <mi xref="p1.1.m1.1.3.1.cmml" id="p1.1.m1.1.3.1"/>
+ <mo xref="p1.1.m1.1.1.cmml" id="p1.1.m1.1.1">*</mo>
+ <mi xref="p1.1.m1.1.2.cmml" id="p1.1.m1.1.2">k</mi>
+ </mrow>
+ <annotation-xml id="p1.1.m1.1b" encoding="MathML-Content">
+ <apply xref="p1.1.m1.1.3" id="p1.1.m1.1.3.cmml">
+ <times xref="p1.1.m1.1.1" id="p1.1.m1.1.1.cmml"/>
+ <csymbol xref="p1.1.m1.1.3.1" id="p1.1.m1.1.3.1.cmml" cd="latexml">absent</csymbol>
+ <ci xref="p1.1.m1.1.2" id="p1.1.m1.1.2.cmml">ð‘˜</ci>
+ </apply>
+ </annotation-xml>
+ <annotation id="p1.1.m1.1c" encoding="application/x-tex">*k</annotation>
+ </semantics>
+</math> on <math id="7662179593224293755" display="inline" alttext="?_{n}" class="ltx_Math" xmlns="http://www.w3.org/1998/Math/MathML">
+ <semantics id="p1.1.m1.1a">
+ <msub xref="p1.1.m1.1.3.cmml" id="p1.1.m1.1.3">
+ <mi xref="p1.1.m1.1.1.cmml" id="p1.1.m1.1.1" mathvariant="normal">?</mi>
+ <mi xref="p1.1.m1.1.2.1.cmml" id="p1.1.m1.1.2.1">n</mi>
+ </msub>
+ <annotation-xml id="p1.1.m1.1b" encoding="MathML-Content">
+ <apply xref="p1.1.m1.1.3" id="p1.1.m1.1.3.cmml">
+ <csymbol id="p1.1.m1.1.3.1.cmml" cd="ambiguous">subscript</csymbol>
+ <ci xref="p1.1.m1.1.1" id="p1.1.m1.1.1.cmml">?</ci>
+ <ci xref="p1.1.m1.1.2.1" id="p1.1.m1.1.2.1.cmml">ð‘›</ci>
+ </apply>
+ </annotation-xml>
+ <annotation id="p1.1.m1.1c" encoding="application/x-tex">?_{n}</annotation>
+ </semantics>
+</math> that are defined by <code class="code">E</code><math id="-3240071424451291920" display="inline" alttext="(n)^{{*k}}=" class="ltx_Math" xmlns="http://www.w3.org/1998/Math/MathML">
+ <semantics id="p1.1.m1.1a">
+ <mrow xref="p1.1.m1.1.6.cmml" id="p1.1.m1.1.6">
+ <msup xref="p1.1.m1.1.6.1.cmml" id="p1.1.m1.1.6.1">
+ <mrow id="p1.1.m1.1.6.1.2">
+ <mo id="p1.1.m1.1.1" stretchy="false">(</mo>
+ <mi xref="p1.1.m1.1.2.cmml" id="p1.1.m1.1.2">n</mi>
+ <mo id="p1.1.m1.1.3" stretchy="false">)</mo>
+ </mrow>
+ <mrow xref="p1.1.m1.1.4.1.cmml" id="p1.1.m1.1.4.1">
+ <mi xref="p1.1.m1.1.4.1.3.cmml" id="p1.1.m1.1.4.1.3"/>
+ <mo xref="p1.1.m1.1.4.1.1.cmml" id="p1.1.m1.1.4.1.1">*</mo>
+ <mi xref="p1.1.m1.1.4.1.2.cmml" id="p1.1.m1.1.4.1.2">k</mi>
+ </mrow>
+ </msup>
+ <mo xref="p1.1.m1.1.5.cmml" id="p1.1.m1.1.5">=</mo>
+ <mi xref="p1.1.m1.1.6.2.cmml" id="p1.1.m1.1.6.2"/>
+ </mrow>
+ <annotation-xml id="p1.1.m1.1b" encoding="MathML-Content">
+ <apply xref="p1.1.m1.1.6" id="p1.1.m1.1.6.cmml">
+ <eq xref="p1.1.m1.1.5" id="p1.1.m1.1.5.cmml"/>
+ <apply xref="p1.1.m1.1.6.1" id="p1.1.m1.1.6.1.cmml">
+ <csymbol id="p1.1.m1.1.6.1.1.cmml" cd="ambiguous">superscript</csymbol>
+ <ci xref="p1.1.m1.1.2" id="p1.1.m1.1.2.cmml">ð‘›</ci>
+ <apply xref="p1.1.m1.1.4.1" id="p1.1.m1.1.4.1.cmml">
+ <times xref="p1.1.m1.1.4.1.1" id="p1.1.m1.1.4.1.1.cmml"/>
+ <csymbol xref="p1.1.m1.1.4.1.3" id="p1.1.m1.1.4.1.3.cmml" cd="latexml">absent</csymbol>
+ <ci xref="p1.1.m1.1.4.1.2" id="p1.1.m1.1.4.1.2.cmml">ð‘˜</ci>
+ </apply>
+ </apply>
+ <csymbol xref="p1.1.m1.1.6.2" id="p1.1.m1.1.6.2.cmml" cd="latexml">absent</csymbol>
+ </apply>
+ </annotation-xml>
+ <annotation id="p1.1.m1.1c" encoding="application/x-tex">(n)^{{*k}}=</annotation>
+ </semantics>
+</math><code class="code">E</code><math id="-7031209855940056214" display="inline" alttext="(n)^{k}" class="ltx_Math" xmlns="http://www.w3.org/1998/Math/MathML">
+ <semantics id="p1.1.m1.1a">
+ <msup xref="p1.1.m1.1.5.cmml" id="p1.1.m1.1.5">
+ <mrow id="p1.1.m1.1.5.2">
+ <mo id="p1.1.m1.1.1" stretchy="false">(</mo>
+ <mi xref="p1.1.m1.1.2.cmml" id="p1.1.m1.1.2">n</mi>
+ <mo id="p1.1.m1.1.3" stretchy="false">)</mo>
+ </mrow>
+ <mi xref="p1.1.m1.1.4.1.cmml" id="p1.1.m1.1.4.1">k</mi>
+ </msup>
+ <annotation-xml id="p1.1.m1.1b" encoding="MathML-Content">
+ <apply xref="p1.1.m1.1.5" id="p1.1.m1.1.5.cmml">
+ <csymbol id="p1.1.m1.1.5.1.cmml" cd="ambiguous">superscript</csymbol>
+ <ci xref="p1.1.m1.1.2" id="p1.1.m1.1.2.cmml">ð‘›</ci>
+ <ci xref="p1.1.m1.1.4.1" id="p1.1.m1.1.4.1.cmml">ð‘˜</ci>
+ </apply>
+ </annotation-xml>
+ <annotation id="p1.1.m1.1c" encoding="application/x-tex">(n)^{k}</annotation>
+ </semantics>
+</math>, where <math id="2685378551882520261" display="inline" alttext="1\leq k&lt;n" class="ltx_Math" xmlns="http://www.w3.org/1998/Math/MathML">
+ <semantics id="p1.1.m1.1a">
+ <mrow xref="p1.1.m1.1.9.cmml" id="p1.1.m1.1.9">
+ <mn xref="p1.1.m1.1.1.cmml" id="p1.1.m1.1.1">1</mn>
+ <mo xref="p1.1.m1.1.2.cmml" id="p1.1.m1.1.2">≤</mo>
+ <mrow xref="p1.1.m1.1.9.1.1.cmml" id="p1.1.m1.1.9.1">
+ <mrow xref="p1.1.m1.1.9.1.1.cmml" id="p1.1.m1.1.9.1.2">
+ <mi xref="p1.1.m1.1.3.cmml" id="p1.1.m1.1.3">k</mi>
+ <mo xref="p1.1.m1.1.9.1.2.1.cmml" id="p1.1.m1.1.9.1.2.1">â¢</mo>
+ <mi xref="p1.1.m1.1.4.cmml" id="p1.1.m1.1.4" mathvariant="normal">&</mi>
+ <mo xref="p1.1.m1.1.9.1.2.1.cmml" id="p1.1.m1.1.9.1.2.1a">â¢</mo>
+ <mi xref="p1.1.m1.1.5.cmml" id="p1.1.m1.1.5">l</mi>
+ <mo xref="p1.1.m1.1.9.1.2.1.cmml" id="p1.1.m1.1.9.1.2.1b">â¢</mo>
+ <mi xref="p1.1.m1.1.6.cmml" id="p1.1.m1.1.6">t</mi>
+ </mrow>
+ <mo id="p1.1.m1.1.7">;</mo>
+ <mi xref="p1.1.m1.1.8.cmml" id="p1.1.m1.1.8">n</mi>
+ </mrow>
+ </mrow>
+ <annotation-xml id="p1.1.m1.1b" encoding="MathML-Content">
+ <apply xref="p1.1.m1.1.9" id="p1.1.m1.1.9.cmml">
+ <leq xref="p1.1.m1.1.2" id="p1.1.m1.1.2.cmml"/>
+ <cn xref="p1.1.m1.1.1" id="p1.1.m1.1.1.cmml" type="integer">1</cn>
+ <list xref="p1.1.m1.1.9.1" id="p1.1.m1.1.9.1.1.cmml">
+ <apply xref="p1.1.m1.1.9.1" id="p1.1.m1.1.9.1.2.cmml">
+ <times xref="p1.1.m1.1.9.1.2.1" id="p1.1.m1.1.9.1.2.1.cmml"/>
+ <ci xref="p1.1.m1.1.3" id="p1.1.m1.1.3.cmml">ð‘˜</ci>
+ <ci xref="p1.1.m1.1.4" id="p1.1.m1.1.4.cmml">&</ci>
+ <ci xref="p1.1.m1.1.5" id="p1.1.m1.1.5.cmml">ð‘™</ci>
+ <ci xref="p1.1.m1.1.6" id="p1.1.m1.1.6.cmml">ð‘¡</ci>
+ </apply>
+ <ci xref="p1.1.m1.1.8" id="p1.1.m1.1.8.cmml">ð‘›</ci>
+ </list>
+ </apply>
+ </annotation-xml>
+ <annotation id="p1.1.m1.1c" encoding="application/x-tex">1\leq k&lt;n</annotation>
+ </semantics>
+</math> and <code class="code">Gcd</code><math id="351920092830706621" display="inline" alttext="(n,k)=1" class="ltx_Math" xmlns="http://www.w3.org/1998/Math/MathML">
+ <semantics id="p1.1.m1.1a">
+ <mrow xref="p1.1.m1.1.8.cmml" id="p1.1.m1.1.8">
+ <mrow xref="p1.1.m1.1.8.1.1.cmml" id="p1.1.m1.1.8.1">
+ <mo id="p1.1.m1.1.1" stretchy="false">(</mo>
+ <mi xref="p1.1.m1.1.2.cmml" id="p1.1.m1.1.2">n</mi>
+ <mo id="p1.1.m1.1.3">,</mo>
+ <mi xref="p1.1.m1.1.4.cmml" id="p1.1.m1.1.4">k</mi>
+ <mo id="p1.1.m1.1.5" stretchy="false">)</mo>
+ </mrow>
+ <mo xref="p1.1.m1.1.6.cmml" id="p1.1.m1.1.6">=</mo>
+ <mn xref="p1.1.m1.1.7.cmml" id="p1.1.m1.1.7">1</mn>
+ </mrow>
+ <annotation-xml id="p1.1.m1.1b" encoding="MathML-Content">
+ <apply xref="p1.1.m1.1.8" id="p1.1.m1.1.8.cmml">
+ <eq xref="p1.1.m1.1.6" id="p1.1.m1.1.6.cmml"/>
+ <interval xref="p1.1.m1.1.8.1" id="p1.1.m1.1.8.1.1.cmml" closure="open">
+ <ci xref="p1.1.m1.1.2" id="p1.1.m1.1.2.cmml">ð‘›</ci>
+ <ci xref="p1.1.m1.1.4" id="p1.1.m1.1.4.cmml">ð‘˜</ci>
+ </interval>
+ <cn xref="p1.1.m1.1.7" id="p1.1.m1.1.7.cmml" type="integer">1</cn>
+ </apply>
+ </annotation-xml>
+ <annotation id="p1.1.m1.1c" encoding="application/x-tex">(n,k)=1</annotation>
+ </semantics>
+</math> hold (see <code class="func">GaloisCyc</code> (<a shape="rect" href="chap18_mj.html#X79EE9097783128C4"><span class="RefLink">18.5-1</span></a>)). Note that this action is <em>not</em> equal to exponentiation of cyclotomics, i.e., for general cyclotomics <math id="-54504977581366215" display="inline" alttext="z" class="ltx_Math" xmlns="http://www.w3.org/1998/Math/MathML">
+ <semantics id="p1.1.m1.1a">
+ <mi xref="p1.1.m1.1.1.cmml" id="p1.1.m1.1.1">z</mi>
+ <annotation-xml id="p1.1.m1.1b" encoding="MathML-Content">
+ <ci xref="p1.1.m1.1.1" id="p1.1.m1.1.1.cmml">ð‘§</ci>
+ </annotation-xml>
+ <annotation id="p1.1.m1.1c" encoding="application/x-tex">z</annotation>
+ </semantics>
+</math>, <math id="1610307170891981444" display="inline" alttext="z^{{*k}}" class="ltx_Math" xmlns="http://www.w3.org/1998/Math/MathML">
+ <semantics id="p1.1.m1.1a">
+ <msup xref="p1.1.m1.1.3.cmml" id="p1.1.m1.1.3">
+ <mi xref="p1.1.m1.1.1.cmml" id="p1.1.m1.1.1">z</mi>
+ <mrow xref="p1.1.m1.1.2.1.cmml" id="p1.1.m1.1.2.1">
+ <mi xref="p1.1.m1.1.2.1.3.cmml" id="p1.1.m1.1.2.1.3"/>
+ <mo xref="p1.1.m1.1.2.1.1.cmml" id="p1.1.m1.1.2.1.1">*</mo>
+ <mi xref="p1.1.m1.1.2.1.2.cmml" id="p1.1.m1.1.2.1.2">k</mi>
+ </mrow>
+ </msup>
+ <annotation-xml id="p1.1.m1.1b" encoding="MathML-Content">
+ <apply xref="p1.1.m1.1.3" id="p1.1.m1.1.3.cmml">
+ <csymbol id="p1.1.m1.1.3.1.cmml" cd="ambiguous">superscript</csymbol>
+ <ci xref="p1.1.m1.1.1" id="p1.1.m1.1.1.cmml">ð‘§</ci>
+ <apply xref="p1.1.m1.1.2.1" id="p1.1.m1.1.2.1.cmml">
+ <times xref="p1.1.m1.1.2.1.1" id="p1.1.m1.1.2.1.1.cmml"/>
+ <csymbol xref="p1.1.m1.1.2.1.3" id="p1.1.m1.1.2.1.3.cmml" cd="latexml">absent</csymbol>
+ <ci xref="p1.1.m1.1.2.1.2" id="p1.1.m1.1.2.1.2.cmml">ð‘˜</ci>
+ </apply>
+ </apply>
+ </annotation-xml>
+ <annotation id="p1.1.m1.1c" encoding="application/x-tex">z^{{*k}}</annotation>
+ </semantics>
+</math> is different from <math id="5745742262325779948" display="inline" alttext="z^{k}" class="ltx_Math" xmlns="http://www.w3.org/1998/Math/MathML">
+ <semantics id="p1.1.m1.1a">
+ <msup xref="p1.1.m1.1.3.cmml" id="p1.1.m1.1.3">
+ <mi xref="p1.1.m1.1.1.cmml" id="p1.1.m1.1.1">z</mi>
+ <mi xref="p1.1.m1.1.2.1.cmml" id="p1.1.m1.1.2.1">k</mi>
+ </msup>
+ <annotation-xml id="p1.1.m1.1b" encoding="MathML-Content">
+ <apply xref="p1.1.m1.1.3" id="p1.1.m1.1.3.cmml">
+ <csymbol id="p1.1.m1.1.3.1.cmml" cd="ambiguous">superscript</csymbol>
+ <ci xref="p1.1.m1.1.1" id="p1.1.m1.1.1.cmml">ð‘§</ci>
+ <ci xref="p1.1.m1.1.2.1" id="p1.1.m1.1.2.1.cmml">ð‘˜</ci>
+ </apply>
+ </annotation-xml>
+ <annotation id="p1.1.m1.1c" encoding="application/x-tex">z^{k}</annotation>
+ </semantics>
+</math>.</p><p xmlns="http://www.w3.org/1999/xhtml">(In <strong class="pkg">GAP</strong>, the image of a cyclotomic <math id="-6251623302427837955" display="inline" alttext="z" class="ltx_Math" xmlns="http://www.w3.org/1998/Math/MathML">
+ <semantics id="p1.1.m1.1a">
+ <mi xref="p1.1.m1.1.1.cmml" id="p1.1.m1.1.1">z</mi>
+ <annotation-xml id="p1.1.m1.1b" encoding="MathML-Content">
+ <ci xref="p1.1.m1.1.1" id="p1.1.m1.1.1.cmml">ð‘§</ci>
+ </annotation-xml>
+ <annotation id="p1.1.m1.1c" encoding="application/x-tex">z</annotation>
+ </semantics>
+</math> under <math id="3460160945755555129" display="inline" alttext="*k" class="ltx_Math" xmlns="http://www.w3.org/1998/Math/MathML">
+ <semantics id="p1.1.m1.1a">
+ <mrow xref="p1.1.m1.1.3.cmml" id="p1.1.m1.1.3">
+ <mi xref="p1.1.m1.1.3.1.cmml" id="p1.1.m1.1.3.1"/>
+ <mo xref="p1.1.m1.1.1.cmml" id="p1.1.m1.1.1">*</mo>
+ <mi xref="p1.1.m1.1.2.cmml" id="p1.1.m1.1.2">k</mi>
+ </mrow>
+ <annotation-xml id="p1.1.m1.1b" encoding="MathML-Content">
+ <apply xref="p1.1.m1.1.3" id="p1.1.m1.1.3.cmml">
+ <times xref="p1.1.m1.1.1" id="p1.1.m1.1.1.cmml"/>
+ <csymbol xref="p1.1.m1.1.3.1" id="p1.1.m1.1.3.1.cmml" cd="latexml">absent</csymbol>
+ <ci xref="p1.1.m1.1.2" id="p1.1.m1.1.2.cmml">ð‘˜</ci>
+ </apply>
+ </annotation-xml>
+ <annotation id="p1.1.m1.1c" encoding="application/x-tex">*k</annotation>
+ </semantics>
+</math> can be computed as <code class="code">GaloisCyc( </code><math id="7378789492007022656" display="inline" alttext="z,k" class="ltx_Math" xmlns="http://www.w3.org/1998/Math/MathML">
+ <semantics id="p1.1.m1.1a">
+ <mrow xref="p1.1.m1.1.4.1.cmml" id="p1.1.m1.1.4">
+ <mi xref="p1.1.m1.1.1.cmml" id="p1.1.m1.1.1">z</mi>
+ <mo id="p1.1.m1.1.2">,</mo>
+ <mi xref="p1.1.m1.1.3.cmml" id="p1.1.m1.1.3">k</mi>
+ </mrow>
+ <annotation-xml id="p1.1.m1.1b" encoding="MathML-Content">
+ <list xref="p1.1.m1.1.4" id="p1.1.m1.1.4.1.cmml">
+ <ci xref="p1.1.m1.1.1" id="p1.1.m1.1.1.cmml">ð‘§</ci>
+ <ci xref="p1.1.m1.1.3" id="p1.1.m1.1.3.cmml">ð‘˜</ci>
+ </list>
+ </annotation-xml>
+ <annotation id="p1.1.m1.1c" encoding="application/x-tex">z,k</annotation>
+ </semantics>
+</math><code class="code"> )</code>.)</p><div class="example" xmlns="http://www.w3.org/1999/xhtml"><pre xml:space="preserve">
+<span class="GAPprompt">gap></span> <span class="GAPinput">( E(5) + E(5)^4 )^2; GaloisCyc( E(5) + E(5)^4, 2 );</span>
+-2*E(5)-E(5)^2-E(5)^3-2*E(5)^4
+E(5)^2+E(5)^3
+</pre></div><p xmlns="http://www.w3.org/1999/xhtml">For <code class="code">Gcd</code><math id="2429103618119238021" display="inline" alttext="(n,k)\neq 1" class="ltx_Math" xmlns="http://www.w3.org/1998/Math/MathML">
+ <semantics id="p1.1.m1.1a">
+ <mrow xref="p1.1.m1.1.8.cmml" id="p1.1.m1.1.8">
+ <mrow xref="p1.1.m1.1.8.1.1.cmml" id="p1.1.m1.1.8.1">
+ <mo id="p1.1.m1.1.1" stretchy="false">(</mo>
+ <mi xref="p1.1.m1.1.2.cmml" id="p1.1.m1.1.2">n</mi>
+ <mo id="p1.1.m1.1.3">,</mo>
+ <mi xref="p1.1.m1.1.4.cmml" id="p1.1.m1.1.4">k</mi>
+ <mo id="p1.1.m1.1.5" stretchy="false">)</mo>
+ </mrow>
+ <mo xref="p1.1.m1.1.6.cmml" id="p1.1.m1.1.6">≠</mo>
+ <mn xref="p1.1.m1.1.7.cmml" id="p1.1.m1.1.7">1</mn>
+ </mrow>
+ <annotation-xml id="p1.1.m1.1b" encoding="MathML-Content">
+ <apply xref="p1.1.m1.1.8" id="p1.1.m1.1.8.cmml">
+ <neq xref="p1.1.m1.1.6" id="p1.1.m1.1.6.cmml"/>
+ <interval xref="p1.1.m1.1.8.1" id="p1.1.m1.1.8.1.1.cmml" closure="open">
+ <ci xref="p1.1.m1.1.2" id="p1.1.m1.1.2.cmml">ð‘›</ci>
+ <ci xref="p1.1.m1.1.4" id="p1.1.m1.1.4.cmml">ð‘˜</ci>
+ </interval>
+ <cn xref="p1.1.m1.1.7" id="p1.1.m1.1.7.cmml" type="integer">1</cn>
+ </apply>
+ </annotation-xml>
+ <annotation id="p1.1.m1.1c" encoding="application/x-tex">(n,k)\neq 1</annotation>
+ </semantics>
+</math>, the map <code class="code">E</code><math id="-17755269626048527" display="inline" alttext="(n)\mapsto" class="ltx_Math" xmlns="http://www.w3.org/1998/Math/MathML">
+ <semantics id="p1.1.m1.1a">
+ <mrow xref="p1.1.m1.1.5.cmml" id="p1.1.m1.1.5">
+ <mrow id="p1.1.m1.1.5.1">
+ <mo id="p1.1.m1.1.1" stretchy="false">(</mo>
+ <mi xref="p1.1.m1.1.2.cmml" id="p1.1.m1.1.2">n</mi>
+ <mo id="p1.1.m1.1.3" stretchy="false">)</mo>
+ </mrow>
+ <mo xref="p1.1.m1.1.4.cmml" id="p1.1.m1.1.4">↦</mo>
+ <mi xref="p1.1.m1.1.5.2.cmml" id="p1.1.m1.1.5.2"/>
+ </mrow>
+ <annotation-xml id="p1.1.m1.1b" encoding="MathML-Content">
+ <apply xref="p1.1.m1.1.5" id="p1.1.m1.1.5.cmml">
+ <csymbol xref="p1.1.m1.1.4" id="p1.1.m1.1.4.cmml" cd="latexml">maps-to</csymbol>
+ <ci xref="p1.1.m1.1.2" id="p1.1.m1.1.2.cmml">ð‘›</ci>
+ <csymbol xref="p1.1.m1.1.5.2" id="p1.1.m1.1.5.2.cmml" cd="latexml">absent</csymbol>
+ </apply>
+ </annotation-xml>
+ <annotation id="p1.1.m1.1c" encoding="application/x-tex">(n)\mapsto</annotation>
+ </semantics>
+</math> <code class="code">E</code><math id="7936211141160956024" display="inline" alttext="(n)^{k}" class="ltx_Math" xmlns="http://www.w3.org/1998/Math/MathML">
+ <semantics id="p1.1.m1.1a">
+ <msup xref="p1.1.m1.1.5.cmml" id="p1.1.m1.1.5">
+ <mrow id="p1.1.m1.1.5.2">
+ <mo id="p1.1.m1.1.1" stretchy="false">(</mo>
+ <mi xref="p1.1.m1.1.2.cmml" id="p1.1.m1.1.2">n</mi>
+ <mo id="p1.1.m1.1.3" stretchy="false">)</mo>
+ </mrow>
+ <mi xref="p1.1.m1.1.4.1.cmml" id="p1.1.m1.1.4.1">k</mi>
+ </msup>
+ <annotation-xml id="p1.1.m1.1b" encoding="MathML-Content">
+ <apply xref="p1.1.m1.1.5" id="p1.1.m1.1.5.cmml">
+ <csymbol id="p1.1.m1.1.5.1.cmml" cd="ambiguous">superscript</csymbol>
+ <ci xref="p1.1.m1.1.2" id="p1.1.m1.1.2.cmml">ð‘›</ci>
+ <ci xref="p1.1.m1.1.4.1" id="p1.1.m1.1.4.1.cmml">ð‘˜</ci>
+ </apply>
+ </annotation-xml>
+ <annotation id="p1.1.m1.1c" encoding="application/x-tex">(n)^{k}</annotation>
+ </semantics>
+</math> does <em>not</em> define a field automorphism of <math id="4830811453257139423" display="inline" alttext="?_{n}" class="ltx_Math" xmlns="http://www.w3.org/1998/Math/MathML">
+ <semantics id="p1.1.m1.1a">
+ <msub xref="p1.1.m1.1.3.cmml" id="p1.1.m1.1.3">
+ <mi xref="p1.1.m1.1.1.cmml" id="p1.1.m1.1.1" mathvariant="normal">?</mi>
+ <mi xref="p1.1.m1.1.2.1.cmml" id="p1.1.m1.1.2.1">n</mi>
+ </msub>
+ <annotation-xml id="p1.1.m1.1b" encoding="MathML-Content">
+ <apply xref="p1.1.m1.1.3" id="p1.1.m1.1.3.cmml">
+ <csymbol id="p1.1.m1.1.3.1.cmml" cd="ambiguous">subscript</csymbol>
+ <ci xref="p1.1.m1.1.1" id="p1.1.m1.1.1.cmml">?</ci>
+ <ci xref="p1.1.m1.1.2.1" id="p1.1.m1.1.2.1.cmml">ð‘›</ci>
+ </apply>
+ </annotation-xml>
+ <annotation id="p1.1.m1.1c" encoding="application/x-tex">?_{n}</annotation>
+ </semantics>
+</math> but only a <math id="-5246741972512659832" display="inline" alttext="?" class="ltx_Math" xmlns="http://www.w3.org/1998/Math/MathML">
+ <semantics id="p1.1.m1.1a">
+ <mi xref="p1.1.m1.1.1.cmml" id="p1.1.m1.1.1" mathvariant="normal">?</mi>
+ <annotation-xml id="p1.1.m1.1b" encoding="MathML-Content">
+ <ci xref="p1.1.m1.1.1" id="p1.1.m1.1.1.cmml">?</ci>
+ </annotation-xml>
+ <annotation id="p1.1.m1.1c" encoding="application/x-tex">?</annotation>
+ </semantics>
+</math>-linear map.</p><div class="example" xmlns="http://www.w3.org/1999/xhtml"><pre xml:space="preserve">
+<span class="GAPprompt">gap></span> <span class="GAPinput">GaloisCyc( E(5)+E(5)^4, 5 ); GaloisCyc( ( E(5)+E(5)^4 )^2, 5 );</span>
+2
+-6
+</pre></div> </body>
+ </html>
\ No newline at end of file
diff --git a/export/mwscrawler/13.html b/export/mwscrawler/13.html
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--- /dev/null
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@@ -0,0 +1,195 @@
+<html>
+ <head>
+ <title>60.3-2 LenstraBase</title>
+ <meta name="url" content="http://www.gap-system.org/Manuals/doc/ref/TODO#X87DB9C2C858B722A"></meta>
+ </head>
+ <body> <h5 xmlns="http://www.w3.org/1999/xhtml">60.3-2 LenstraBase</h5><div class="func" xmlns="http://www.w3.org/1999/xhtml"><table width="100%" class="func"><tbody><tr><td colspan="1" rowspan="1" class="tdleft"><code class="func">‣ LenstraBase</code>( <var class="Arg">n</var>, <var class="Arg">stabilizer</var>, <var class="Arg">super</var>, <var class="Arg">m</var> )</td><td colspan="1" rowspan="1" class="tdright">( function )</td></tr></tbody></table></div><p xmlns="http://www.w3.org/1999/xhtml">Let <var class="Arg">n</var> and <var class="Arg">m</var> be positive integers such that <var class="Arg">m</var> divides <var class="Arg">n</var>, <var class="Arg">stabilizer</var> be a list of prime residues modulo <var class="Arg">n</var>, which describes a subfield of the <var class="Arg">n</var>-th cyclotomic field (see <code class="func">GaloisStabilizer</code> (<a shape="rect" href="chap60_mj.html#X87E7313D8070B9CC"><span class="RefLink">60.2-5</span></a>)), and <var class="Arg">super</var> be a list representing a supergroup of the group given by <var class="Arg">stabilizer</var>.</p><p xmlns="http://www.w3.org/1999/xhtml"><code class="func">LenstraBase</code> returns a list <math id="1208490429264458309" display="inline" alttext="[b_{1},b_{2},\ldots,b_{k}]" class="ltx_Math" xmlns="http://www.w3.org/1998/Math/MathML">
+ <semantics id="p1.1.m1.1a">
+ <mrow xref="p1.1.m1.1.13.1.cmml" id="p1.1.m1.1.13">
+ <mo id="p1.1.m1.1.1" stretchy="false">[</mo>
+ <msub xref="p1.1.m1.1.13.1.cmml" id="p1.1.m1.1.13.2">
+ <mi xref="p1.1.m1.1.2.cmml" id="p1.1.m1.1.2">b</mi>
+ <mn xref="p1.1.m1.1.3.1.cmml" id="p1.1.m1.1.3.1">1</mn>
+ </msub>
+ <mo id="p1.1.m1.1.4">,</mo>
+ <msub xref="p1.1.m1.1.13.1.cmml" id="p1.1.m1.1.13.3">
+ <mi xref="p1.1.m1.1.5.cmml" id="p1.1.m1.1.5">b</mi>
+ <mn xref="p1.1.m1.1.6.1.cmml" id="p1.1.m1.1.6.1">2</mn>
+ </msub>
+ <mo id="p1.1.m1.1.7">,</mo>
+ <mi xref="p1.1.m1.1.8.cmml" id="p1.1.m1.1.8" mathvariant="normal">…</mi>
+ <mo id="p1.1.m1.1.9">,</mo>
+ <msub xref="p1.1.m1.1.13.1.cmml" id="p1.1.m1.1.13.4">
+ <mi xref="p1.1.m1.1.10.cmml" id="p1.1.m1.1.10">b</mi>
+ <mi xref="p1.1.m1.1.11.1.cmml" id="p1.1.m1.1.11.1">k</mi>
+ </msub>
+ <mo id="p1.1.m1.1.12" stretchy="false">]</mo>
+ </mrow>
+ <annotation-xml id="p1.1.m1.1b" encoding="MathML-Content">
+ <list xref="p1.1.m1.1.13" id="p1.1.m1.1.13.1.cmml">
+ <apply xref="p1.1.m1.1.13" id="p1.1.m1.1.13.2.cmml">
+ <csymbol id="p1.1.m1.1.13.2.1.cmml" cd="ambiguous">subscript</csymbol>
+ <ci xref="p1.1.m1.1.2" id="p1.1.m1.1.2.cmml">ð‘</ci>
+ <cn xref="p1.1.m1.1.3.1" id="p1.1.m1.1.3.1.cmml" type="integer">1</cn>
+ </apply>
+ <apply xref="p1.1.m1.1.13" id="p1.1.m1.1.13.3.cmml">
+ <csymbol id="p1.1.m1.1.13.3.1.cmml" cd="ambiguous">subscript</csymbol>
+ <ci xref="p1.1.m1.1.5" id="p1.1.m1.1.5.cmml">ð‘</ci>
+ <cn xref="p1.1.m1.1.6.1" id="p1.1.m1.1.6.1.cmml" type="integer">2</cn>
+ </apply>
+ <ci xref="p1.1.m1.1.8" id="p1.1.m1.1.8.cmml">…</ci>
+ <apply xref="p1.1.m1.1.13" id="p1.1.m1.1.13.4.cmml">
+ <csymbol id="p1.1.m1.1.13.4.1.cmml" cd="ambiguous">subscript</csymbol>
+ <ci xref="p1.1.m1.1.10" id="p1.1.m1.1.10.cmml">ð‘</ci>
+ <ci xref="p1.1.m1.1.11.1" id="p1.1.m1.1.11.1.cmml">ð‘˜</ci>
+ </apply>
+ </list>
+ </annotation-xml>
+ <annotation id="p1.1.m1.1c" encoding="application/x-tex">[b_{1},b_{2},\ldots,b_{k}]</annotation>
+ </semantics>
+</math> of lists, each <math id="-3701198119998990270" display="inline" alttext="b_{i}" class="ltx_Math" xmlns="http://www.w3.org/1998/Math/MathML">
+ <semantics id="p1.1.m1.1a">
+ <msub xref="p1.1.m1.1.3.cmml" id="p1.1.m1.1.3">
+ <mi xref="p1.1.m1.1.1.cmml" id="p1.1.m1.1.1">b</mi>
+ <mi xref="p1.1.m1.1.2.1.cmml" id="p1.1.m1.1.2.1">i</mi>
+ </msub>
+ <annotation-xml id="p1.1.m1.1b" encoding="MathML-Content">
+ <apply xref="p1.1.m1.1.3" id="p1.1.m1.1.3.cmml">
+ <csymbol id="p1.1.m1.1.3.1.cmml" cd="ambiguous">subscript</csymbol>
+ <ci xref="p1.1.m1.1.1" id="p1.1.m1.1.1.cmml">ð‘</ci>
+ <ci xref="p1.1.m1.1.2.1" id="p1.1.m1.1.2.1.cmml">ð‘–</ci>
+ </apply>
+ </annotation-xml>
+ <annotation id="p1.1.m1.1c" encoding="application/x-tex">b_{i}</annotation>
+ </semantics>
+</math> consisting of integers such that the elements <math id="-8645456467319532217" display="inline" alttext="\sum_{{j\in b_{i}}}" class="ltx_Math" xmlns="http://www.w3.org/1998/Math/MathML">
+ <semantics id="p1.1.m1.1a">
+ <msub xref="p1.1.m1.1.3.cmml" id="p1.1.m1.1.3">
+ <mo xref="p1.1.m1.1.1.cmml" id="p1.1.m1.1.1" symmetric="true" largeop="true">∑</mo>
+ <mrow xref="p1.1.m1.1.2.1.cmml" id="p1.1.m1.1.2.1">
+ <mi xref="p1.1.m1.1.2.1.1.cmml" id="p1.1.m1.1.2.1.1">j</mi>
+ <mo xref="p1.1.m1.1.2.1.2.cmml" id="p1.1.m1.1.2.1.2">∈</mo>
+ <msub xref="p1.1.m1.1.2.1.5.cmml" id="p1.1.m1.1.2.1.5">
+ <mi xref="p1.1.m1.1.2.1.3.cmml" id="p1.1.m1.1.2.1.3">b</mi>
+ <mi xref="p1.1.m1.1.2.1.4.1.cmml" id="p1.1.m1.1.2.1.4.1">i</mi>
+ </msub>
+ </mrow>
+ </msub>
+ <annotation-xml id="p1.1.m1.1b" encoding="MathML-Content">
+ <apply xref="p1.1.m1.1.3" id="p1.1.m1.1.3.cmml">
+ <csymbol id="p1.1.m1.1.3.1.cmml" cd="ambiguous">subscript</csymbol>
+ <sum xref="p1.1.m1.1.1" id="p1.1.m1.1.1.cmml"/>
+ <apply xref="p1.1.m1.1.2.1" id="p1.1.m1.1.2.1.cmml">
+ <in xref="p1.1.m1.1.2.1.2" id="p1.1.m1.1.2.1.2.cmml"/>
+ <ci xref="p1.1.m1.1.2.1.1" id="p1.1.m1.1.2.1.1.cmml">ð‘—</ci>
+ <apply xref="p1.1.m1.1.2.1.5" id="p1.1.m1.1.2.1.5.cmml">
+ <csymbol id="p1.1.m1.1.2.1.5.1.cmml" cd="ambiguous">subscript</csymbol>
+ <ci xref="p1.1.m1.1.2.1.3" id="p1.1.m1.1.2.1.3.cmml">ð‘</ci>
+ <ci xref="p1.1.m1.1.2.1.4.1" id="p1.1.m1.1.2.1.4.1.cmml">ð‘–</ci>
+ </apply>
+ </apply>
+ </apply>
+ </annotation-xml>
+ <annotation id="p1.1.m1.1c" encoding="application/x-tex">\sum_{{j\in b_{i}}}</annotation>
+ </semantics>
+</math><code class="code">E(n)</code><math id="-5417637940190866429" display="inline" alttext="{}^{j}" class="ltx_Math" xmlns="http://www.w3.org/1998/Math/MathML">
+ <semantics id="p1.1.m1.1a">
+ <msup xref="p1.1.m1.1.1.cmml" id="p1.1.m1.1.1">
+ <mi xref="p1.1.m1.1.1.cmml" id="p1.1.m1.1.1a"/>
+ <mi xref="p1.1.m1.1.1.1.cmml" id="p1.1.m1.1.1.1">j</mi>
+ </msup>
+ <annotation-xml id="p1.1.m1.1b" encoding="MathML-Content">
+ <apply xref="p1.1.m1.1.1" id="p1.1.m1.1.1.cmml">
+ <ci xref="p1.1.m1.1.1.1" id="p1.1.m1.1.1.1.cmml">ð‘—</ci>
+ </apply>
+ </annotation-xml>
+ <annotation id="p1.1.m1.1c" encoding="application/x-tex">{}^{j}</annotation>
+ </semantics>
+</math> form a basis of the abelian number field <code class="code">NF( <var class="Arg">n</var>, <var class="Arg">stabilizer</var> )</code>, as a vector space over the <var class="Arg">m</var>-th cyclotomic field (see <code class="func">AbelianNumberField</code> (<a shape="rect" href="chap60_mj.html#X80E5AD028143E11E"><span class="RefLink">60.1-2</span></a>)).</p><p xmlns="http://www.w3.org/1999/xhtml">This basis is an integral basis, that is, exactly the integral elements in <code class="code">NF( <var class="Arg">n</var>, <var class="Arg">stabilizer</var> )</code> have integral coefficients. (For details about this basis, see <a shape="rect" href="chapBib_mj.html#biBBre97">[Bre97]</a>.)</p><p xmlns="http://www.w3.org/1999/xhtml">If possible then the result is chosen such that the group described by <var class="Arg">super</var> acts on it, consistently with the action of <var class="Arg">stabilizer</var>, i.e., each orbit of <var class="Arg">super</var> is a union of orbits of <var class="Arg">stabilizer</var>. (A usual case is <var class="Arg">super</var><code class="code"> = </code><var class="Arg">stabilizer</var>, so there is no additional condition.</p><p xmlns="http://www.w3.org/1999/xhtml"><em>Note:</em> The <math id="2032741142945267551" display="inline" alttext="b_{i}" class="ltx_Math" xmlns="http://www.w3.org/1998/Math/MathML">
+ <semantics id="p1.1.m1.1a">
+ <msub xref="p1.1.m1.1.3.cmml" id="p1.1.m1.1.3">
+ <mi xref="p1.1.m1.1.1.cmml" id="p1.1.m1.1.1">b</mi>
+ <mi xref="p1.1.m1.1.2.1.cmml" id="p1.1.m1.1.2.1">i</mi>
+ </msub>
+ <annotation-xml id="p1.1.m1.1b" encoding="MathML-Content">
+ <apply xref="p1.1.m1.1.3" id="p1.1.m1.1.3.cmml">
+ <csymbol id="p1.1.m1.1.3.1.cmml" cd="ambiguous">subscript</csymbol>
+ <ci xref="p1.1.m1.1.1" id="p1.1.m1.1.1.cmml">ð‘</ci>
+ <ci xref="p1.1.m1.1.2.1" id="p1.1.m1.1.2.1.cmml">ð‘–</ci>
+ </apply>
+ </annotation-xml>
+ <annotation id="p1.1.m1.1c" encoding="application/x-tex">b_{i}</annotation>
+ </semantics>
+</math> are in general not sets, since for <code class="code"><var class="Arg">stabilizer</var> = <var class="Arg">super</var></code>, the first entry is always an element of <code class="code">ZumbroichBase( <var class="Arg">n</var>, <var class="Arg">m</var> )</code>; this property is used by <code class="func">NF</code> (<a shape="rect" href="chap60_mj.html#X80E5AD028143E11E"><span class="RefLink">60.1-2</span></a>) and <code class="func">Coefficients</code> (<a shape="rect" href="chap61_mj.html#X80B32F667BF6AFD8"><span class="RefLink">61.6-3</span></a>) (see <a shape="rect" href="chap60_mj.html#X7D2421AC8491D2BE"><span class="RefLink">60.3</span></a>).</p><p xmlns="http://www.w3.org/1999/xhtml"><var class="Arg">stabilizer</var> must not contain the stabilizer of a proper cyclotomic subfield of the <var class="Arg">n</var>-th cyclotomic field, i.e., the result must describe a basis for a field with conductor <var class="Arg">n</var>.</p><div class="example" xmlns="http://www.w3.org/1999/xhtml"><pre xml:space="preserve">
+<span class="GAPprompt">gap></span> <span class="GAPinput">LenstraBase( 24, [ 1, 19 ], [ 1, 19 ], 1 );</span>
+[ [ 1, 19 ], [ 8 ], [ 11, 17 ], [ 16 ] ]
+<span class="GAPprompt">gap></span> <span class="GAPinput">LenstraBase( 24, [ 1, 19 ], [ 1, 5, 19, 23 ], 1 );</span>
+[ [ 1, 19 ], [ 5, 23 ], [ 8 ], [ 16 ] ]
+<span class="GAPprompt">gap></span> <span class="GAPinput">LenstraBase( 15, [ 1, 4 ], PrimeResidues( 15 ), 1 );</span>
+[ [ 1, 4 ], [ 2, 8 ], [ 7, 13 ], [ 11, 14 ] ]
+</pre></div><p xmlns="http://www.w3.org/1999/xhtml">The first two results describe two bases of the field <math id="-6590770517438568572" display="inline" alttext="?_{3}(\sqrt{{6}})" class="ltx_Math" xmlns="http://www.w3.org/1998/Math/MathML">
+ <semantics id="p1.1.m1.1a">
+ <mrow xref="p1.1.m1.1.6.cmml" id="p1.1.m1.1.6">
+ <msub xref="p1.1.m1.1.6.2.cmml" id="p1.1.m1.1.6.2">
+ <mi xref="p1.1.m1.1.1.cmml" id="p1.1.m1.1.1" mathvariant="normal">?</mi>
+ <mn xref="p1.1.m1.1.2.1.cmml" id="p1.1.m1.1.2.1">3</mn>
+ </msub>
+ <mo xref="p1.1.m1.1.6.1.cmml" id="p1.1.m1.1.6.1">â¢</mo>
+ <mrow xref="p1.1.m1.1.4.cmml" id="p1.1.m1.1.6.3">
+ <mo id="p1.1.m1.1.3" stretchy="false">(</mo>
+ <msqrt xref="p1.1.m1.1.4.cmml" id="p1.1.m1.1.4">
+ <mn xref="p1.1.m1.1.4.2.cmml" id="p1.1.m1.1.4.2">6</mn>
+ </msqrt>
+ <mo id="p1.1.m1.1.5" stretchy="false">)</mo>
+ </mrow>
+ </mrow>
+ <annotation-xml id="p1.1.m1.1b" encoding="MathML-Content">
+ <apply xref="p1.1.m1.1.6" id="p1.1.m1.1.6.cmml">
+ <times xref="p1.1.m1.1.6.1" id="p1.1.m1.1.6.1.cmml"/>
+ <apply xref="p1.1.m1.1.6.2" id="p1.1.m1.1.6.2.cmml">
+ <csymbol id="p1.1.m1.1.6.2.1.cmml" cd="ambiguous">subscript</csymbol>
+ <ci xref="p1.1.m1.1.1" id="p1.1.m1.1.1.cmml">?</ci>
+ <cn xref="p1.1.m1.1.2.1" id="p1.1.m1.1.2.1.cmml" type="integer">3</cn>
+ </apply>
+ <apply xref="p1.1.m1.1.6.3" id="p1.1.m1.1.4.cmml">
+ <root id="p1.1.m1.1.4a.cmml"/>
+ <cn xref="p1.1.m1.1.4.2" id="p1.1.m1.1.4.2.cmml" type="integer">6</cn>
+ </apply>
+ </apply>
+ </annotation-xml>
+ <annotation id="p1.1.m1.1c" encoding="application/x-tex">?_{3}(\sqrt{{6}})</annotation>
+ </semantics>
+</math>, the third result describes a normal basis of <math id="-6848104539316450458" display="inline" alttext="?_{3}(\sqrt{{5}})" class="ltx_Math" xmlns="http://www.w3.org/1998/Math/MathML">
+ <semantics id="p1.1.m1.1a">
+ <mrow xref="p1.1.m1.1.6.cmml" id="p1.1.m1.1.6">
+ <msub xref="p1.1.m1.1.6.2.cmml" id="p1.1.m1.1.6.2">
+ <mi xref="p1.1.m1.1.1.cmml" id="p1.1.m1.1.1" mathvariant="normal">?</mi>
+ <mn xref="p1.1.m1.1.2.1.cmml" id="p1.1.m1.1.2.1">3</mn>
+ </msub>
+ <mo xref="p1.1.m1.1.6.1.cmml" id="p1.1.m1.1.6.1">â¢</mo>
+ <mrow xref="p1.1.m1.1.4.cmml" id="p1.1.m1.1.6.3">
+ <mo id="p1.1.m1.1.3" stretchy="false">(</mo>
+ <msqrt xref="p1.1.m1.1.4.cmml" id="p1.1.m1.1.4">
+ <mn xref="p1.1.m1.1.4.2.cmml" id="p1.1.m1.1.4.2">5</mn>
+ </msqrt>
+ <mo id="p1.1.m1.1.5" stretchy="false">)</mo>
+ </mrow>
+ </mrow>
+ <annotation-xml id="p1.1.m1.1b" encoding="MathML-Content">
+ <apply xref="p1.1.m1.1.6" id="p1.1.m1.1.6.cmml">
+ <times xref="p1.1.m1.1.6.1" id="p1.1.m1.1.6.1.cmml"/>
+ <apply xref="p1.1.m1.1.6.2" id="p1.1.m1.1.6.2.cmml">
+ <csymbol id="p1.1.m1.1.6.2.1.cmml" cd="ambiguous">subscript</csymbol>
+ <ci xref="p1.1.m1.1.1" id="p1.1.m1.1.1.cmml">?</ci>
+ <cn xref="p1.1.m1.1.2.1" id="p1.1.m1.1.2.1.cmml" type="integer">3</cn>
+ </apply>
+ <apply xref="p1.1.m1.1.6.3" id="p1.1.m1.1.4.cmml">
+ <root id="p1.1.m1.1.4a.cmml"/>
+ <cn xref="p1.1.m1.1.4.2" id="p1.1.m1.1.4.2.cmml" type="integer">5</cn>
+ </apply>
+ </apply>
+ </annotation-xml>
+ <annotation id="p1.1.m1.1c" encoding="application/x-tex">?_{3}(\sqrt{{5}})</annotation>
+ </semantics>
+</math>.</p> </body>
+ </html>
\ No newline at end of file
diff --git a/export/mwscrawler/14.html b/export/mwscrawler/14.html
new file mode 100644
index 0000000..e69a180
--- /dev/null
+++ b/export/mwscrawler/14.html
@@ -0,0 +1,888 @@
+<html>
+ <head>
+ <title>60.3-1 ZumbroichBase</title>
+ <meta name="url" content="http://www.gap-system.org/Manuals/doc/ref/TODO#X7F52BEA0862E06F2"></meta>
+ </head>
+ <body> <h5 xmlns="http://www.w3.org/1999/xhtml">60.3-1 ZumbroichBase</h5><div class="func" xmlns="http://www.w3.org/1999/xhtml"><table width="100%" class="func"><tbody><tr><td colspan="1" rowspan="1" class="tdleft"><code class="func">‣ ZumbroichBase</code>( <var class="Arg">n</var>, <var class="Arg">m</var> )</td><td colspan="1" rowspan="1" class="tdright">( function )</td></tr></tbody></table></div><p xmlns="http://www.w3.org/1999/xhtml">Let <var class="Arg">n</var> and <var class="Arg">m</var> be positive integers, such that <var class="Arg">m</var> divides <var class="Arg">n</var>. <code class="func">ZumbroichBase</code> returns the set of exponents <math id="796425337723754006" display="inline" alttext="i" class="ltx_Math" xmlns="http://www.w3.org/1998/Math/MathML">
+ <semantics id="p1.1.m1.1a">
+ <mi xref="p1.1.m1.1.1.cmml" id="p1.1.m1.1.1">i</mi>
+ <annotation-xml id="p1.1.m1.1b" encoding="MathML-Content">
+ <ci xref="p1.1.m1.1.1" id="p1.1.m1.1.1.cmml">ð‘–</ci>
+ </annotation-xml>
+ <annotation id="p1.1.m1.1c" encoding="application/x-tex">i</annotation>
+ </semantics>
+</math> for which <code class="code">E(<var class="Arg">n</var>)^</code><math id="4810311857378126922" display="inline" alttext="i" class="ltx_Math" xmlns="http://www.w3.org/1998/Math/MathML">
+ <semantics id="p1.1.m1.1a">
+ <mi xref="p1.1.m1.1.1.cmml" id="p1.1.m1.1.1">i</mi>
+ <annotation-xml id="p1.1.m1.1b" encoding="MathML-Content">
+ <ci xref="p1.1.m1.1.1" id="p1.1.m1.1.1.cmml">ð‘–</ci>
+ </annotation-xml>
+ <annotation id="p1.1.m1.1c" encoding="application/x-tex">i</annotation>
+ </semantics>
+</math> belongs to the (generalized) Zumbroich basis of the cyclotomic field <math id="6180988263455099748" display="inline" alttext="?_{n}" class="ltx_Math" xmlns="http://www.w3.org/1998/Math/MathML">
+ <semantics id="p1.1.m1.1a">
+ <msub xref="p1.1.m1.1.3.cmml" id="p1.1.m1.1.3">
+ <mi xref="p1.1.m1.1.1.cmml" id="p1.1.m1.1.1" mathvariant="normal">?</mi>
+ <mi xref="p1.1.m1.1.2.1.cmml" id="p1.1.m1.1.2.1">n</mi>
+ </msub>
+ <annotation-xml id="p1.1.m1.1b" encoding="MathML-Content">
+ <apply xref="p1.1.m1.1.3" id="p1.1.m1.1.3.cmml">
+ <csymbol id="p1.1.m1.1.3.1.cmml" cd="ambiguous">subscript</csymbol>
+ <ci xref="p1.1.m1.1.1" id="p1.1.m1.1.1.cmml">?</ci>
+ <ci xref="p1.1.m1.1.2.1" id="p1.1.m1.1.2.1.cmml">ð‘›</ci>
+ </apply>
+ </annotation-xml>
+ <annotation id="p1.1.m1.1c" encoding="application/x-tex">?_{n}</annotation>
+ </semantics>
+</math>, viewed as a vector space over <math id="-2368206857514410044" display="inline" alttext="?_{m}" class="ltx_Math" xmlns="http://www.w3.org/1998/Math/MathML">
+ <semantics id="p1.1.m1.1a">
+ <msub xref="p1.1.m1.1.3.cmml" id="p1.1.m1.1.3">
+ <mi xref="p1.1.m1.1.1.cmml" id="p1.1.m1.1.1" mathvariant="normal">?</mi>
+ <mi xref="p1.1.m1.1.2.1.cmml" id="p1.1.m1.1.2.1">m</mi>
+ </msub>
+ <annotation-xml id="p1.1.m1.1b" encoding="MathML-Content">
+ <apply xref="p1.1.m1.1.3" id="p1.1.m1.1.3.cmml">
+ <csymbol id="p1.1.m1.1.3.1.cmml" cd="ambiguous">subscript</csymbol>
+ <ci xref="p1.1.m1.1.1" id="p1.1.m1.1.1.cmml">?</ci>
+ <ci xref="p1.1.m1.1.2.1" id="p1.1.m1.1.2.1.cmml">ð‘š</ci>
+ </apply>
+ </annotation-xml>
+ <annotation id="p1.1.m1.1c" encoding="application/x-tex">?_{m}</annotation>
+ </semantics>
+</math>.</p><p xmlns="http://www.w3.org/1999/xhtml">This basis is defined as follows. Let <math id="-4433440537360578210" display="inline" alttext="P" class="ltx_Math" xmlns="http://www.w3.org/1998/Math/MathML">
+ <semantics id="p1.1.m1.1a">
+ <mi xref="p1.1.m1.1.1.cmml" id="p1.1.m1.1.1">P</mi>
+ <annotation-xml id="p1.1.m1.1b" encoding="MathML-Content">
+ <ci xref="p1.1.m1.1.1" id="p1.1.m1.1.1.cmml">ð‘ƒ</ci>
+ </annotation-xml>
+ <annotation id="p1.1.m1.1c" encoding="application/x-tex">P</annotation>
+ </semantics>
+</math> denote the set of prime divisors of <var class="Arg">n</var>, <math id="-2231488919819557843" display="inline" alttext="\textit{n}=\prod_{{p\in P}}p^{{\nu_{p}}}" class="ltx_Math" xmlns="http://www.w3.org/1998/Math/MathML">
+ <semantics id="p1.1.m1.1a">
+ <mrow xref="p1.1.m1.1.7.cmml" id="p1.1.m1.1.7">
+ <mtext xref="p1.1.m1.1.1.1.cmml" id="p1.1.m1.1.1.1">ð‘›</mtext>
+ <mo xref="p1.1.m1.1.2.cmml" id="p1.1.m1.1.2">=</mo>
+ <mrow xref="p1.1.m1.1.7.1.cmml" id="p1.1.m1.1.7.1">
+ <msub xref="p1.1.m1.1.7.1.1.cmml" id="p1.1.m1.1.7.1.1">
+ <mo xref="p1.1.m1.1.3.cmml" id="p1.1.m1.1.3" symmetric="true" largeop="true">âˆ</mo>
+ <mrow xref="p1.1.m1.1.4.1.cmml" id="p1.1.m1.1.4.1">
+ <mi xref="p1.1.m1.1.4.1.1.cmml" id="p1.1.m1.1.4.1.1">p</mi>
+ <mo xref="p1.1.m1.1.4.1.2.cmml" id="p1.1.m1.1.4.1.2">∈</mo>
+ <mi xref="p1.1.m1.1.4.1.3.cmml" id="p1.1.m1.1.4.1.3">P</mi>
+ </mrow>
+ </msub>
+ <msup xref="p1.1.m1.1.7.1.2.cmml" id="p1.1.m1.1.7.1.2">
+ <mi xref="p1.1.m1.1.5.cmml" id="p1.1.m1.1.5">p</mi>
+ <msub xref="p1.1.m1.1.6.1.cmml" id="p1.1.m1.1.6.1">
+ <mi xref="p1.1.m1.1.6.1.1.cmml" id="p1.1.m1.1.6.1.1">ν</mi>
+ <mi xref="p1.1.m1.1.6.1.2.1.cmml" id="p1.1.m1.1.6.1.2.1">p</mi>
+ </msub>
+ </msup>
+ </mrow>
+ </mrow>
+ <annotation-xml id="p1.1.m1.1b" encoding="MathML-Content">
+ <apply xref="p1.1.m1.1.7" id="p1.1.m1.1.7.cmml">
+ <eq xref="p1.1.m1.1.2" id="p1.1.m1.1.2.cmml"/>
+ <mtext xref="p1.1.m1.1.1.1" id="p1.1.m1.1.1.1.cmml">
+ <span class="ltx_text ltx_font_italic" xmlns="http://www.w3.org/1999/xhtml">n</span>
+ </mtext>
+ <apply xref="p1.1.m1.1.7.1" id="p1.1.m1.1.7.1.cmml">
+ <apply xref="p1.1.m1.1.7.1.1" id="p1.1.m1.1.7.1.1.cmml">
+ <csymbol id="p1.1.m1.1.7.1.1.1.cmml" cd="ambiguous">subscript</csymbol>
+ <csymbol xref="p1.1.m1.1.3" id="p1.1.m1.1.3.cmml" cd="latexml">product</csymbol>
+ <apply xref="p1.1.m1.1.4.1" id="p1.1.m1.1.4.1.cmml">
+ <in xref="p1.1.m1.1.4.1.2" id="p1.1.m1.1.4.1.2.cmml"/>
+ <ci xref="p1.1.m1.1.4.1.1" id="p1.1.m1.1.4.1.1.cmml">ð‘</ci>
+ <ci xref="p1.1.m1.1.4.1.3" id="p1.1.m1.1.4.1.3.cmml">ð‘ƒ</ci>
+ </apply>
+ </apply>
+ <apply xref="p1.1.m1.1.7.1.2" id="p1.1.m1.1.7.1.2.cmml">
+ <csymbol id="p1.1.m1.1.7.1.2.1.cmml" cd="ambiguous">superscript</csymbol>
+ <ci xref="p1.1.m1.1.5" id="p1.1.m1.1.5.cmml">ð‘</ci>
+ <apply xref="p1.1.m1.1.6.1" id="p1.1.m1.1.6.1.cmml">
+ <csymbol id="p1.1.m1.1.6.1.3.cmml" cd="ambiguous">subscript</csymbol>
+ <ci xref="p1.1.m1.1.6.1.1" id="p1.1.m1.1.6.1.1.cmml">ðœˆ</ci>
+ <ci xref="p1.1.m1.1.6.1.2.1" id="p1.1.m1.1.6.1.2.1.cmml">ð‘</ci>
+ </apply>
+ </apply>
+ </apply>
+ </apply>
+ </annotation-xml>
+ <annotation id="p1.1.m1.1c" encoding="application/x-tex">\textit{n}=\prod_{{p\in P}}p^{{\nu_{p}}}</annotation>
+ </semantics>
+</math>, and <math id="5315807327622227726" display="inline" alttext="\textit{m}=\prod_{{p\in P}}p^{{\mu_{p}}}" class="ltx_Math" xmlns="http://www.w3.org/1998/Math/MathML">
+ <semantics id="p1.1.m1.1a">
+ <mrow xref="p1.1.m1.1.7.cmml" id="p1.1.m1.1.7">
+ <mtext xref="p1.1.m1.1.1.1.cmml" id="p1.1.m1.1.1.1">ð‘š</mtext>
+ <mo xref="p1.1.m1.1.2.cmml" id="p1.1.m1.1.2">=</mo>
+ <mrow xref="p1.1.m1.1.7.1.cmml" id="p1.1.m1.1.7.1">
+ <msub xref="p1.1.m1.1.7.1.1.cmml" id="p1.1.m1.1.7.1.1">
+ <mo xref="p1.1.m1.1.3.cmml" id="p1.1.m1.1.3" symmetric="true" largeop="true">âˆ</mo>
+ <mrow xref="p1.1.m1.1.4.1.cmml" id="p1.1.m1.1.4.1">
+ <mi xref="p1.1.m1.1.4.1.1.cmml" id="p1.1.m1.1.4.1.1">p</mi>
+ <mo xref="p1.1.m1.1.4.1.2.cmml" id="p1.1.m1.1.4.1.2">∈</mo>
+ <mi xref="p1.1.m1.1.4.1.3.cmml" id="p1.1.m1.1.4.1.3">P</mi>
+ </mrow>
+ </msub>
+ <msup xref="p1.1.m1.1.7.1.2.cmml" id="p1.1.m1.1.7.1.2">
+ <mi xref="p1.1.m1.1.5.cmml" id="p1.1.m1.1.5">p</mi>
+ <msub xref="p1.1.m1.1.6.1.cmml" id="p1.1.m1.1.6.1">
+ <mi xref="p1.1.m1.1.6.1.1.cmml" id="p1.1.m1.1.6.1.1">μ</mi>
+ <mi xref="p1.1.m1.1.6.1.2.1.cmml" id="p1.1.m1.1.6.1.2.1">p</mi>
+ </msub>
+ </msup>
+ </mrow>
+ </mrow>
+ <annotation-xml id="p1.1.m1.1b" encoding="MathML-Content">
+ <apply xref="p1.1.m1.1.7" id="p1.1.m1.1.7.cmml">
+ <eq xref="p1.1.m1.1.2" id="p1.1.m1.1.2.cmml"/>
+ <mtext xref="p1.1.m1.1.1.1" id="p1.1.m1.1.1.1.cmml">
+ <span class="ltx_text ltx_font_italic" xmlns="http://www.w3.org/1999/xhtml">m</span>
+ </mtext>
+ <apply xref="p1.1.m1.1.7.1" id="p1.1.m1.1.7.1.cmml">
+ <apply xref="p1.1.m1.1.7.1.1" id="p1.1.m1.1.7.1.1.cmml">
+ <csymbol id="p1.1.m1.1.7.1.1.1.cmml" cd="ambiguous">subscript</csymbol>
+ <csymbol xref="p1.1.m1.1.3" id="p1.1.m1.1.3.cmml" cd="latexml">product</csymbol>
+ <apply xref="p1.1.m1.1.4.1" id="p1.1.m1.1.4.1.cmml">
+ <in xref="p1.1.m1.1.4.1.2" id="p1.1.m1.1.4.1.2.cmml"/>
+ <ci xref="p1.1.m1.1.4.1.1" id="p1.1.m1.1.4.1.1.cmml">ð‘</ci>
+ <ci xref="p1.1.m1.1.4.1.3" id="p1.1.m1.1.4.1.3.cmml">ð‘ƒ</ci>
+ </apply>
+ </apply>
+ <apply xref="p1.1.m1.1.7.1.2" id="p1.1.m1.1.7.1.2.cmml">
+ <csymbol id="p1.1.m1.1.7.1.2.1.cmml" cd="ambiguous">superscript</csymbol>
+ <ci xref="p1.1.m1.1.5" id="p1.1.m1.1.5.cmml">ð‘</ci>
+ <apply xref="p1.1.m1.1.6.1" id="p1.1.m1.1.6.1.cmml">
+ <csymbol id="p1.1.m1.1.6.1.3.cmml" cd="ambiguous">subscript</csymbol>
+ <ci xref="p1.1.m1.1.6.1.1" id="p1.1.m1.1.6.1.1.cmml">ðœ‡</ci>
+ <ci xref="p1.1.m1.1.6.1.2.1" id="p1.1.m1.1.6.1.2.1.cmml">ð‘</ci>
+ </apply>
+ </apply>
+ </apply>
+ </apply>
+ </annotation-xml>
+ <annotation id="p1.1.m1.1c" encoding="application/x-tex">\textit{m}=\prod_{{p\in P}}p^{{\mu_{p}}}</annotation>
+ </semantics>
+</math> with <math id="-432699783012467212" display="inline" alttext="\mu_{p}\leq\nu_{p}" class="ltx_Math" xmlns="http://www.w3.org/1998/Math/MathML">
+ <semantics id="p1.1.m1.1a">
+ <mrow xref="p1.1.m1.1.6.cmml" id="p1.1.m1.1.6">
+ <msub xref="p1.1.m1.1.6.1.cmml" id="p1.1.m1.1.6.1">
+ <mi xref="p1.1.m1.1.1.cmml" id="p1.1.m1.1.1">μ</mi>
+ <mi xref="p1.1.m1.1.2.1.cmml" id="p1.1.m1.1.2.1">p</mi>
+ </msub>
+ <mo xref="p1.1.m1.1.3.cmml" id="p1.1.m1.1.3">≤</mo>
+ <msub xref="p1.1.m1.1.6.2.cmml" id="p1.1.m1.1.6.2">
+ <mi xref="p1.1.m1.1.4.cmml" id="p1.1.m1.1.4">ν</mi>
+ <mi xref="p1.1.m1.1.5.1.cmml" id="p1.1.m1.1.5.1">p</mi>
+ </msub>
+ </mrow>
+ <annotation-xml id="p1.1.m1.1b" encoding="MathML-Content">
+ <apply xref="p1.1.m1.1.6" id="p1.1.m1.1.6.cmml">
+ <leq xref="p1.1.m1.1.3" id="p1.1.m1.1.3.cmml"/>
+ <apply xref="p1.1.m1.1.6.1" id="p1.1.m1.1.6.1.cmml">
+ <csymbol id="p1.1.m1.1.6.1.1.cmml" cd="ambiguous">subscript</csymbol>
+ <ci xref="p1.1.m1.1.1" id="p1.1.m1.1.1.cmml">ðœ‡</ci>
+ <ci xref="p1.1.m1.1.2.1" id="p1.1.m1.1.2.1.cmml">ð‘</ci>
+ </apply>
+ <apply xref="p1.1.m1.1.6.2" id="p1.1.m1.1.6.2.cmml">
+ <csymbol id="p1.1.m1.1.6.2.1.cmml" cd="ambiguous">subscript</csymbol>
+ <ci xref="p1.1.m1.1.4" id="p1.1.m1.1.4.cmml">ðœˆ</ci>
+ <ci xref="p1.1.m1.1.5.1" id="p1.1.m1.1.5.1.cmml">ð‘</ci>
+ </apply>
+ </apply>
+ </annotation-xml>
+ <annotation id="p1.1.m1.1c" encoding="application/x-tex">\mu_{p}\leq\nu_{p}</annotation>
+ </semantics>
+</math>. Let <math id="6322515088010280443" display="inline" alttext="e_{l}=" class="ltx_Math" xmlns="http://www.w3.org/1998/Math/MathML">
+ <semantics id="p1.1.m1.1a">
+ <mrow xref="p1.1.m1.1.4.cmml" id="p1.1.m1.1.4">
+ <msub xref="p1.1.m1.1.4.1.cmml" id="p1.1.m1.1.4.1">
+ <mi xref="p1.1.m1.1.1.cmml" id="p1.1.m1.1.1">e</mi>
+ <mi xref="p1.1.m1.1.2.1.cmml" id="p1.1.m1.1.2.1">l</mi>
+ </msub>
+ <mo xref="p1.1.m1.1.3.cmml" id="p1.1.m1.1.3">=</mo>
+ <mi xref="p1.1.m1.1.4.2.cmml" id="p1.1.m1.1.4.2"/>
+ </mrow>
+ <annotation-xml id="p1.1.m1.1b" encoding="MathML-Content">
+ <apply xref="p1.1.m1.1.4" id="p1.1.m1.1.4.cmml">
+ <eq xref="p1.1.m1.1.3" id="p1.1.m1.1.3.cmml"/>
+ <apply xref="p1.1.m1.1.4.1" id="p1.1.m1.1.4.1.cmml">
+ <csymbol id="p1.1.m1.1.4.1.1.cmml" cd="ambiguous">subscript</csymbol>
+ <ci xref="p1.1.m1.1.1" id="p1.1.m1.1.1.cmml">ð‘’</ci>
+ <ci xref="p1.1.m1.1.2.1" id="p1.1.m1.1.2.1.cmml">ð‘™</ci>
+ </apply>
+ <csymbol xref="p1.1.m1.1.4.2" id="p1.1.m1.1.4.2.cmml" cd="latexml">absent</csymbol>
+ </apply>
+ </annotation-xml>
+ <annotation id="p1.1.m1.1c" encoding="application/x-tex">e_{l}=</annotation>
+ </semantics>
+</math> <code class="code">E</code><math id="-3581860542562778662" display="inline" alttext="(l)" class="ltx_Math" xmlns="http://www.w3.org/1998/Math/MathML">
+ <semantics id="p1.1.m1.1a">
+ <mrow id="p1.1.m1.1.4">
+ <mo id="p1.1.m1.1.1" stretchy="false">(</mo>
+ <mi xref="p1.1.m1.1.2.cmml" id="p1.1.m1.1.2">l</mi>
+ <mo id="p1.1.m1.1.3" stretchy="false">)</mo>
+ </mrow>
+ <annotation-xml id="p1.1.m1.1b" encoding="MathML-Content">
+ <ci xref="p1.1.m1.1.2" id="p1.1.m1.1.2.cmml">ð‘™</ci>
+ </annotation-xml>
+ <annotation id="p1.1.m1.1c" encoding="application/x-tex">(l)</annotation>
+ </semantics>
+</math> for any positive integer <math id="4469849179910767759" display="inline" alttext="l" class="ltx_Math" xmlns="http://www.w3.org/1998/Math/MathML">
+ <semantics id="p1.1.m1.1a">
+ <mi xref="p1.1.m1.1.1.cmml" id="p1.1.m1.1.1">l</mi>
+ <annotation-xml id="p1.1.m1.1b" encoding="MathML-Content">
+ <ci xref="p1.1.m1.1.1" id="p1.1.m1.1.1.cmml">ð‘™</ci>
+ </annotation-xml>
+ <annotation id="p1.1.m1.1c" encoding="application/x-tex">l</annotation>
+ </semantics>
+</math>, and <math id="8455169956076990367" display="inline" alttext="\{e_{{n_{1}}}^{j}\}_{{j\in J}}\otimes\{e_{{n_{2}}}^{k}\}_{{k\in K}}=\{e_{{n_{1%
+}}}^{j}\cdot e_{{n_{2}}}^{k}\}_{{j\in J,k\in K}}" class="ltx_Math" xmlns="http://www.w3.org/1998/Math/MathML">
+ <semantics id="p1.1.m1.1a">
+ <mrow xref="p1.1.m1.1.25.cmml" id="p1.1.m1.1.25">
+ <mrow xref="p1.1.m1.1.25.1.cmml" id="p1.1.m1.1.25.1">
+ <msub xref="p1.1.m1.1.25.1.1.cmml" id="p1.1.m1.1.25.1.1">
+ <mrow xref="p1.1.m1.1.25.1.1.2.1.cmml" id="p1.1.m1.1.25.1.1.2">
+ <mo id="p1.1.m1.1.1" stretchy="false">{</mo>
+ <msubsup xref="p1.1.m1.1.25.1.1.2.1.cmml" id="p1.1.m1.1.25.1.1.2.2">
+ <mi xref="p1.1.m1.1.2.cmml" id="p1.1.m1.1.2">e</mi>
+ <msub xref="p1.1.m1.1.25.1.1.2.1.cmml" id="p1.1.m1.1.3.1">
+ <mi xref="p1.1.m1.1.3.1.1.cmml" id="p1.1.m1.1.3.1.1">n</mi>
+ <mn xref="p1.1.m1.1.3.1.2.1.cmml" id="p1.1.m1.1.3.1.2.1">1</mn>
+ </msub>
+ <mi xref="p1.1.m1.1.4.1.cmml" id="p1.1.m1.1.4.1">j</mi>
+ </msubsup>
+ <mo id="p1.1.m1.1.5" stretchy="false">}</mo>
+ </mrow>
+ <mrow xref="p1.1.m1.1.6.1.cmml" id="p1.1.m1.1.6.1">
+ <mi xref="p1.1.m1.1.6.1.1.cmml" id="p1.1.m1.1.6.1.1">j</mi>
+ <mo xref="p1.1.m1.1.6.1.2.cmml" id="p1.1.m1.1.6.1.2">∈</mo>
+ <mi xref="p1.1.m1.1.6.1.3.cmml" id="p1.1.m1.1.6.1.3">J</mi>
+ </mrow>
+ </msub>
+ <mo xref="p1.1.m1.1.7.cmml" id="p1.1.m1.1.7">⊗</mo>
+ <msub xref="p1.1.m1.1.25.1.2.cmml" id="p1.1.m1.1.25.1.2">
+ <mrow xref="p1.1.m1.1.25.1.2.2.1.cmml" id="p1.1.m1.1.25.1.2.2">
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+ <mn xref="p1.1.m1.1.10.1.2.1.cmml" id="p1.1.m1.1.10.1.2.1">2</mn>
+ </msub>
+ <mi xref="p1.1.m1.1.11.1.cmml" id="p1.1.m1.1.11.1">k</mi>
+ </msubsup>
+ <mo id="p1.1.m1.1.12" stretchy="false">}</mo>
+ </mrow>
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+ </mrow>
+ </msub>
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+ <mi xref="p1.1.m1.1.18.1.cmml" id="p1.1.m1.1.18.1">j</mi>
+ </msubsup>
+ <mo xref="p1.1.m1.1.19.cmml" id="p1.1.m1.1.19">â‹…</mo>
+ <msubsup xref="p1.1.m1.1.25.2.2.1.cmml" id="p1.1.m1.1.25.2.2.2.2">
+ <mi xref="p1.1.m1.1.20.cmml" id="p1.1.m1.1.20">e</mi>
+ <msub xref="p1.1.m1.1.25.2.2.1.cmml" id="p1.1.m1.1.21.1">
+ <mi xref="p1.1.m1.1.21.1.1.cmml" id="p1.1.m1.1.21.1.1">n</mi>
+ <mn xref="p1.1.m1.1.21.1.2.1.cmml" id="p1.1.m1.1.21.1.2.1">2</mn>
+ </msub>
+ <mi xref="p1.1.m1.1.22.1.cmml" id="p1.1.m1.1.22.1">k</mi>
+ </msubsup>
+ </mrow>
+ <mo id="p1.1.m1.1.23" stretchy="false">}</mo>
+ </mrow>
+ <mrow xref="p1.1.m1.1.24.1.8.cmml" id="p1.1.m1.1.24.1">
+ <mrow xref="p1.1.m1.1.24.1.8.cmml" id="p1.1.m1.1.24.1.9">
+ <mi xref="p1.1.m1.1.24.1.1.cmml" id="p1.1.m1.1.24.1.1">j</mi>
+ <mo xref="p1.1.m1.1.24.1.2.cmml" id="p1.1.m1.1.24.1.2">∈</mo>
+ <mi xref="p1.1.m1.1.24.1.3.cmml" id="p1.1.m1.1.24.1.3">J</mi>
+ </mrow>
+ <mo xref="p1.1.m1.1.24.1.8a.cmml" id="p1.1.m1.1.24.1.4">,</mo>
+ <mrow xref="p1.1.m1.1.24.1.8.cmml" id="p1.1.m1.1.24.1.10">
+ <mi xref="p1.1.m1.1.24.1.5.cmml" id="p1.1.m1.1.24.1.5">k</mi>
+ <mo xref="p1.1.m1.1.24.1.6.cmml" id="p1.1.m1.1.24.1.6">∈</mo>
+ <mi xref="p1.1.m1.1.24.1.7.cmml" id="p1.1.m1.1.24.1.7">K</mi>
+ </mrow>
+ </mrow>
+ </msub>
+ </mrow>
+ <annotation-xml id="p1.1.m1.1b" encoding="MathML-Content">
+ <apply xref="p1.1.m1.1.25" id="p1.1.m1.1.25.cmml">
+ <eq xref="p1.1.m1.1.14" id="p1.1.m1.1.14.cmml"/>
+ <apply xref="p1.1.m1.1.25.1" id="p1.1.m1.1.25.1.cmml">
+ <csymbol xref="p1.1.m1.1.7" id="p1.1.m1.1.7.cmml" cd="latexml">tensor-product</csymbol>
+ <apply xref="p1.1.m1.1.25.1.1" id="p1.1.m1.1.25.1.1.cmml">
+ <csymbol id="p1.1.m1.1.25.1.1.1.cmml" cd="ambiguous">subscript</csymbol>
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+ <apply xref="p1.1.m1.1.25.1.1.2" id="p1.1.m1.1.25.1.1.2.2.cmml">
+ <csymbol id="p1.1.m1.1.25.1.1.2.2.1.cmml" cd="ambiguous">superscript</csymbol>
+ <apply xref="p1.1.m1.1.25.1.1.2" id="p1.1.m1.1.25.1.1.2.2.2.cmml">
+ <csymbol id="p1.1.m1.1.25.1.1.2.2.2.1.cmml" cd="ambiguous">subscript</csymbol>
+ <ci xref="p1.1.m1.1.2" id="p1.1.m1.1.2.cmml">ð‘’</ci>
+ <apply xref="p1.1.m1.1.25.1.1.2" id="p1.1.m1.1.3.1.cmml">
+ <csymbol id="p1.1.m1.1.3.1.3.cmml" cd="ambiguous">subscript</csymbol>
+ <ci xref="p1.1.m1.1.3.1.1" id="p1.1.m1.1.3.1.1.cmml">ð‘›</ci>
+ <cn xref="p1.1.m1.1.3.1.2.1" id="p1.1.m1.1.3.1.2.1.cmml" type="integer">1</cn>
+ </apply>
+ </apply>
+ <ci xref="p1.1.m1.1.4.1" id="p1.1.m1.1.4.1.cmml">ð‘—</ci>
+ </apply>
+ </set>
+ <apply xref="p1.1.m1.1.6.1" id="p1.1.m1.1.6.1.cmml">
+ <in xref="p1.1.m1.1.6.1.2" id="p1.1.m1.1.6.1.2.cmml"/>
+ <ci xref="p1.1.m1.1.6.1.1" id="p1.1.m1.1.6.1.1.cmml">ð‘—</ci>
+ <ci xref="p1.1.m1.1.6.1.3" id="p1.1.m1.1.6.1.3.cmml">ð½</ci>
+ </apply>
+ </apply>
+ <apply xref="p1.1.m1.1.25.1.2" id="p1.1.m1.1.25.1.2.cmml">
+ <csymbol id="p1.1.m1.1.25.1.2.1.cmml" cd="ambiguous">subscript</csymbol>
+ <set xref="p1.1.m1.1.25.1.2.2" id="p1.1.m1.1.25.1.2.2.1.cmml">
+ <apply xref="p1.1.m1.1.25.1.2.2" id="p1.1.m1.1.25.1.2.2.2.cmml">
+ <csymbol id="p1.1.m1.1.25.1.2.2.2.1.cmml" cd="ambiguous">superscript</csymbol>
+ <apply xref="p1.1.m1.1.25.1.2.2" id="p1.1.m1.1.25.1.2.2.2.2.cmml">
+ <csymbol id="p1.1.m1.1.25.1.2.2.2.2.1.cmml" cd="ambiguous">subscript</csymbol>
+ <ci xref="p1.1.m1.1.9" id="p1.1.m1.1.9.cmml">ð‘’</ci>
+ <apply xref="p1.1.m1.1.25.1.2.2" id="p1.1.m1.1.10.1.cmml">
+ <csymbol id="p1.1.m1.1.10.1.3.cmml" cd="ambiguous">subscript</csymbol>
+ <ci xref="p1.1.m1.1.10.1.1" id="p1.1.m1.1.10.1.1.cmml">ð‘›</ci>
+ <cn xref="p1.1.m1.1.10.1.2.1" id="p1.1.m1.1.10.1.2.1.cmml" type="integer">2</cn>
+ </apply>
+ </apply>
+ <ci xref="p1.1.m1.1.11.1" id="p1.1.m1.1.11.1.cmml">ð‘˜</ci>
+ </apply>
+ </set>
+ <apply xref="p1.1.m1.1.13.1" id="p1.1.m1.1.13.1.cmml">
+ <in xref="p1.1.m1.1.13.1.2" id="p1.1.m1.1.13.1.2.cmml"/>
+ <ci xref="p1.1.m1.1.13.1.1" id="p1.1.m1.1.13.1.1.cmml">ð‘˜</ci>
+ <ci xref="p1.1.m1.1.13.1.3" id="p1.1.m1.1.13.1.3.cmml">ð¾</ci>
+ </apply>
+ </apply>
+ </apply>
+ <apply xref="p1.1.m1.1.25.2" id="p1.1.m1.1.25.2.cmml">
+ <csymbol id="p1.1.m1.1.25.2.1.cmml" cd="ambiguous">subscript</csymbol>
+ <set xref="p1.1.m1.1.25.2.2" id="p1.1.m1.1.25.2.2.1.cmml">
+ <apply xref="p1.1.m1.1.25.2.2" id="p1.1.m1.1.25.2.2.2.cmml">
+ <ci xref="p1.1.m1.1.19" id="p1.1.m1.1.19.cmml">â‹…</ci>
+ <apply xref="p1.1.m1.1.25.2.2" id="p1.1.m1.1.25.2.2.2.1.cmml">
+ <csymbol id="p1.1.m1.1.25.2.2.2.1.1.cmml" cd="ambiguous">superscript</csymbol>
+ <apply xref="p1.1.m1.1.25.2.2" id="p1.1.m1.1.25.2.2.2.1.2.cmml">
+ <csymbol id="p1.1.m1.1.25.2.2.2.1.2.1.cmml" cd="ambiguous">subscript</csymbol>
+ <ci xref="p1.1.m1.1.16" id="p1.1.m1.1.16.cmml">ð‘’</ci>
+ <apply xref="p1.1.m1.1.25.2.2" id="p1.1.m1.1.17.1.cmml">
+ <csymbol id="p1.1.m1.1.17.1.3.cmml" cd="ambiguous">subscript</csymbol>
+ <ci xref="p1.1.m1.1.17.1.1" id="p1.1.m1.1.17.1.1.cmml">ð‘›</ci>
+ <cn xref="p1.1.m1.1.17.1.2.1" id="p1.1.m1.1.17.1.2.1.cmml" type="integer">1</cn>
+ </apply>
+ </apply>
+ <ci xref="p1.1.m1.1.18.1" id="p1.1.m1.1.18.1.cmml">ð‘—</ci>
+ </apply>
+ <apply xref="p1.1.m1.1.25.2.2" id="p1.1.m1.1.25.2.2.2.2.cmml">
+ <csymbol id="p1.1.m1.1.25.2.2.2.2.1.cmml" cd="ambiguous">superscript</csymbol>
+ <apply xref="p1.1.m1.1.25.2.2" id="p1.1.m1.1.25.2.2.2.2.2.cmml">
+ <csymbol id="p1.1.m1.1.25.2.2.2.2.2.1.cmml" cd="ambiguous">subscript</csymbol>
+ <ci xref="p1.1.m1.1.20" id="p1.1.m1.1.20.cmml">ð‘’</ci>
+ <apply xref="p1.1.m1.1.25.2.2" id="p1.1.m1.1.21.1.cmml">
+ <csymbol id="p1.1.m1.1.21.1.3.cmml" cd="ambiguous">subscript</csymbol>
+ <ci xref="p1.1.m1.1.21.1.1" id="p1.1.m1.1.21.1.1.cmml">ð‘›</ci>
+ <cn xref="p1.1.m1.1.21.1.2.1" id="p1.1.m1.1.21.1.2.1.cmml" type="integer">2</cn>
+ </apply>
+ </apply>
+ <ci xref="p1.1.m1.1.22.1" id="p1.1.m1.1.22.1.cmml">ð‘˜</ci>
+ </apply>
+ </apply>
+ </set>
+ <apply xref="p1.1.m1.1.24.1" id="p1.1.m1.1.24.1.8.cmml">
+ <csymbol xref="p1.1.m1.1.24.1.4" id="p1.1.m1.1.24.1.8a.cmml" cd="ambiguous">formulae-sequence</csymbol>
+ <apply xref="p1.1.m1.1.24.1" id="p1.1.m1.1.24.1.9.cmml">
+ <in xref="p1.1.m1.1.24.1.2" id="p1.1.m1.1.24.1.2.cmml"/>
+ <ci xref="p1.1.m1.1.24.1.1" id="p1.1.m1.1.24.1.1.cmml">ð‘—</ci>
+ <ci xref="p1.1.m1.1.24.1.3" id="p1.1.m1.1.24.1.3.cmml">ð½</ci>
+ </apply>
+ <apply xref="p1.1.m1.1.24.1" id="p1.1.m1.1.24.1.10.cmml">
+ <in xref="p1.1.m1.1.24.1.6" id="p1.1.m1.1.24.1.6.cmml"/>
+ <ci xref="p1.1.m1.1.24.1.5" id="p1.1.m1.1.24.1.5.cmml">ð‘˜</ci>
+ <ci xref="p1.1.m1.1.24.1.7" id="p1.1.m1.1.24.1.7.cmml">ð¾</ci>
+ </apply>
+ </apply>
+ </apply>
+ </apply>
+ </annotation-xml>
+ <annotation id="p1.1.m1.1c" encoding="application/x-tex">\{e_{{n_{1}}}^{j}\}_{{j\in J}}\otimes\{e_{{n_{2}}}^{k}\}_{{k\in K}}=\{e_{{n_{1%
+}}}^{j}\cdot e_{{n_{2}}}^{k}\}_{{j\in J,k\in K}}</annotation>
+ </semantics>
+</math>.</p><p xmlns="http://www.w3.org/1999/xhtml">Then the basis is</p><p class="center" xmlns="http://www.w3.org/1999/xhtml">\[
+B_{{n,m}} = \bigotimes_{{p \in P}}
+ \bigotimes_{{k = \mu_p}}^{{\nu_p-1}}
+ \{ e_{{p^{{k+1}}}}^j \}_{{j \in J_{{k,p}}}}
+\]</p><p xmlns="http://www.w3.org/1999/xhtml">where <math id="2511610486146337041" display="inline" alttext="J_{{k,p}}=" class="ltx_Math" xmlns="http://www.w3.org/1998/Math/MathML">
+ <semantics id="p1.1.m1.1a">
+ <mrow xref="p1.1.m1.1.4.cmml" id="p1.1.m1.1.4">
+ <msub xref="p1.1.m1.1.4.1.cmml" id="p1.1.m1.1.4.1">
+ <mi xref="p1.1.m1.1.1.cmml" id="p1.1.m1.1.1">J</mi>
+ <mrow xref="p1.1.m1.1.2.1.4.cmml" id="p1.1.m1.1.2.1">
+ <mi xref="p1.1.m1.1.2.1.1.cmml" id="p1.1.m1.1.2.1.1">k</mi>
+ <mo id="p1.1.m1.1.2.1.2">,</mo>
+ <mi xref="p1.1.m1.1.2.1.3.cmml" id="p1.1.m1.1.2.1.3">p</mi>
+ </mrow>
+ </msub>
+ <mo xref="p1.1.m1.1.3.cmml" id="p1.1.m1.1.3">=</mo>
+ <mi xref="p1.1.m1.1.4.2.cmml" id="p1.1.m1.1.4.2"/>
+ </mrow>
+ <annotation-xml id="p1.1.m1.1b" encoding="MathML-Content">
+ <apply xref="p1.1.m1.1.4" id="p1.1.m1.1.4.cmml">
+ <eq xref="p1.1.m1.1.3" id="p1.1.m1.1.3.cmml"/>
+ <apply xref="p1.1.m1.1.4.1" id="p1.1.m1.1.4.1.cmml">
+ <csymbol id="p1.1.m1.1.4.1.1.cmml" cd="ambiguous">subscript</csymbol>
+ <ci xref="p1.1.m1.1.1" id="p1.1.m1.1.1.cmml">ð½</ci>
+ <list xref="p1.1.m1.1.2.1" id="p1.1.m1.1.2.1.4.cmml">
+ <ci xref="p1.1.m1.1.2.1.1" id="p1.1.m1.1.2.1.1.cmml">ð‘˜</ci>
+ <ci xref="p1.1.m1.1.2.1.3" id="p1.1.m1.1.2.1.3.cmml">ð‘</ci>
+ </list>
+ </apply>
+ <csymbol xref="p1.1.m1.1.4.2" id="p1.1.m1.1.4.2.cmml" cd="latexml">absent</csymbol>
+ </apply>
+ </annotation-xml>
+ <annotation id="p1.1.m1.1c" encoding="application/x-tex">J_{{k,p}}=</annotation>
+ </semantics>
+</math></p><div class="pcenter" xmlns="http://www.w3.org/1999/xhtml"><table class="GAPDocTablenoborder"><tbody><tr><td colspan="1" rowspan="1" class="tdleft"><math id="-2031640155713888939" display="inline" alttext="\{0\}" class="ltx_Math" xmlns="http://www.w3.org/1998/Math/MathML">
+ <semantics id="p1.1.m1.1a">
+ <mrow xref="p1.1.m1.1.4.1.cmml" id="p1.1.m1.1.4">
+ <mo id="p1.1.m1.1.1" stretchy="false">{</mo>
+ <mn xref="p1.1.m1.1.2.cmml" id="p1.1.m1.1.2">0</mn>
+ <mo id="p1.1.m1.1.3" stretchy="false">}</mo>
+ </mrow>
+ <annotation-xml id="p1.1.m1.1b" encoding="MathML-Content">
+ <set xref="p1.1.m1.1.4" id="p1.1.m1.1.4.1.cmml">
+ <cn xref="p1.1.m1.1.2" id="p1.1.m1.1.2.cmml" type="integer">0</cn>
+ </set>
+ </annotation-xml>
+ <annotation id="p1.1.m1.1c" encoding="application/x-tex">\{0\}</annotation>
+ </semantics>
+</math></td><td colspan="1" rowspan="1" class="tdcenter">;</td><td colspan="1" rowspan="1" class="tdleft"><math id="-6464892599761167823" display="inline" alttext="k=0,p=2" class="ltx_Math" xmlns="http://www.w3.org/1998/Math/MathML">
+ <semantics id="p1.1.m1.1a">
+ <mrow xref="p1.1.m1.1.8.1.cmml" id="p1.1.m1.1.8">
+ <mrow xref="p1.1.m1.1.8.1.cmml" id="p1.1.m1.1.8.2">
+ <mi xref="p1.1.m1.1.1.cmml" id="p1.1.m1.1.1">k</mi>
+ <mo xref="p1.1.m1.1.2.cmml" id="p1.1.m1.1.2">=</mo>
+ <mn xref="p1.1.m1.1.3.cmml" id="p1.1.m1.1.3">0</mn>
+ </mrow>
+ <mo xref="p1.1.m1.1.8.1a.cmml" id="p1.1.m1.1.4">,</mo>
+ <mrow xref="p1.1.m1.1.8.1.cmml" id="p1.1.m1.1.8.3">
+ <mi xref="p1.1.m1.1.5.cmml" id="p1.1.m1.1.5">p</mi>
+ <mo xref="p1.1.m1.1.6.cmml" id="p1.1.m1.1.6">=</mo>
+ <mn xref="p1.1.m1.1.7.cmml" id="p1.1.m1.1.7">2</mn>
+ </mrow>
+ </mrow>
+ <annotation-xml id="p1.1.m1.1b" encoding="MathML-Content">
+ <apply xref="p1.1.m1.1.8" id="p1.1.m1.1.8.1.cmml">
+ <csymbol xref="p1.1.m1.1.4" id="p1.1.m1.1.8.1a.cmml" cd="ambiguous">formulae-sequence</csymbol>
+ <apply xref="p1.1.m1.1.8" id="p1.1.m1.1.8.2.cmml">
+ <eq xref="p1.1.m1.1.2" id="p1.1.m1.1.2.cmml"/>
+ <ci xref="p1.1.m1.1.1" id="p1.1.m1.1.1.cmml">ð‘˜</ci>
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+ <csymbol id="p1.1.m1.1.3.1.cmml" cd="ambiguous">subscript</csymbol>
+ <ci xref="p1.1.m1.1.1" id="p1.1.m1.1.1.cmml">ðµ</ci>
+ <list xref="p1.1.m1.1.2.1" id="p1.1.m1.1.2.1.4.cmml">
+ <ci xref="p1.1.m1.1.2.1.1" id="p1.1.m1.1.2.1.1.cmml">ð‘›</ci>
+ <cn xref="p1.1.m1.1.2.1.3" id="p1.1.m1.1.2.1.3.cmml" type="integer">1</cn>
+ </list>
+ </apply>
+ </annotation-xml>
+ <annotation id="p1.1.m1.1c" encoding="application/x-tex">B_{{n,1}}</annotation>
+ </semantics>
+</math> is equal to the basis of <math id="504732168073337218" display="inline" alttext="?_{n}" class="ltx_Math" xmlns="http://www.w3.org/1998/Math/MathML">
+ <semantics id="p1.1.m1.1a">
+ <msub xref="p1.1.m1.1.3.cmml" id="p1.1.m1.1.3">
+ <mi xref="p1.1.m1.1.1.cmml" id="p1.1.m1.1.1" mathvariant="normal">?</mi>
+ <mi xref="p1.1.m1.1.2.1.cmml" id="p1.1.m1.1.2.1">n</mi>
+ </msub>
+ <annotation-xml id="p1.1.m1.1b" encoding="MathML-Content">
+ <apply xref="p1.1.m1.1.3" id="p1.1.m1.1.3.cmml">
+ <csymbol id="p1.1.m1.1.3.1.cmml" cd="ambiguous">subscript</csymbol>
+ <ci xref="p1.1.m1.1.1" id="p1.1.m1.1.1.cmml">?</ci>
+ <ci xref="p1.1.m1.1.2.1" id="p1.1.m1.1.2.1.cmml">ð‘›</ci>
+ </apply>
+ </annotation-xml>
+ <annotation id="p1.1.m1.1c" encoding="application/x-tex">?_{n}</annotation>
+ </semantics>
+</math> over the rationals which is introduced in <a shape="rect" href="chapBib_mj.html#biBZum89">[Zum89]</a>. Also the conversion of arbitrary sums of roots of unity into its basis representation, and the reduction to the minimal cyclotomic field are described in this thesis. (Note that the notation here is slightly different from that there.)</p><p xmlns="http://www.w3.org/1999/xhtml"><math id="4598898233162721820" display="inline" alttext="B_{{n,m}}" class="ltx_Math" xmlns="http://www.w3.org/1998/Math/MathML">
+ <semantics id="p1.1.m1.1a">
+ <msub xref="p1.1.m1.1.3.cmml" id="p1.1.m1.1.3">
+ <mi xref="p1.1.m1.1.1.cmml" id="p1.1.m1.1.1">B</mi>
+ <mrow xref="p1.1.m1.1.2.1.4.cmml" id="p1.1.m1.1.2.1">
+ <mi xref="p1.1.m1.1.2.1.1.cmml" id="p1.1.m1.1.2.1.1">n</mi>
+ <mo id="p1.1.m1.1.2.1.2">,</mo>
+ <mi xref="p1.1.m1.1.2.1.3.cmml" id="p1.1.m1.1.2.1.3">m</mi>
+ </mrow>
+ </msub>
+ <annotation-xml id="p1.1.m1.1b" encoding="MathML-Content">
+ <apply xref="p1.1.m1.1.3" id="p1.1.m1.1.3.cmml">
+ <csymbol id="p1.1.m1.1.3.1.cmml" cd="ambiguous">subscript</csymbol>
+ <ci xref="p1.1.m1.1.1" id="p1.1.m1.1.1.cmml">ðµ</ci>
+ <list xref="p1.1.m1.1.2.1" id="p1.1.m1.1.2.1.4.cmml">
+ <ci xref="p1.1.m1.1.2.1.1" id="p1.1.m1.1.2.1.1.cmml">ð‘›</ci>
+ <ci xref="p1.1.m1.1.2.1.3" id="p1.1.m1.1.2.1.3.cmml">ð‘š</ci>
+ </list>
+ </apply>
+ </annotation-xml>
+ <annotation id="p1.1.m1.1c" encoding="application/x-tex">B_{{n,m}}</annotation>
+ </semantics>
+</math> consists of roots of unity, it is an integral basis (that is, exactly the integral elements in <math id="-2895289708463143363" display="inline" alttext="?_{n}" class="ltx_Math" xmlns="http://www.w3.org/1998/Math/MathML">
+ <semantics id="p1.1.m1.1a">
+ <msub xref="p1.1.m1.1.3.cmml" id="p1.1.m1.1.3">
+ <mi xref="p1.1.m1.1.1.cmml" id="p1.1.m1.1.1" mathvariant="normal">?</mi>
+ <mi xref="p1.1.m1.1.2.1.cmml" id="p1.1.m1.1.2.1">n</mi>
+ </msub>
+ <annotation-xml id="p1.1.m1.1b" encoding="MathML-Content">
+ <apply xref="p1.1.m1.1.3" id="p1.1.m1.1.3.cmml">
+ <csymbol id="p1.1.m1.1.3.1.cmml" cd="ambiguous">subscript</csymbol>
+ <ci xref="p1.1.m1.1.1" id="p1.1.m1.1.1.cmml">?</ci>
+ <ci xref="p1.1.m1.1.2.1" id="p1.1.m1.1.2.1.cmml">ð‘›</ci>
+ </apply>
+ </annotation-xml>
+ <annotation id="p1.1.m1.1c" encoding="application/x-tex">?_{n}</annotation>
+ </semantics>
+</math> have integral coefficients w.r.t. <math id="1938984797233526720" display="inline" alttext="B_{{n,m}}" class="ltx_Math" xmlns="http://www.w3.org/1998/Math/MathML">
+ <semantics id="p1.1.m1.1a">
+ <msub xref="p1.1.m1.1.3.cmml" id="p1.1.m1.1.3">
+ <mi xref="p1.1.m1.1.1.cmml" id="p1.1.m1.1.1">B</mi>
+ <mrow xref="p1.1.m1.1.2.1.4.cmml" id="p1.1.m1.1.2.1">
+ <mi xref="p1.1.m1.1.2.1.1.cmml" id="p1.1.m1.1.2.1.1">n</mi>
+ <mo id="p1.1.m1.1.2.1.2">,</mo>
+ <mi xref="p1.1.m1.1.2.1.3.cmml" id="p1.1.m1.1.2.1.3">m</mi>
+ </mrow>
+ </msub>
+ <annotation-xml id="p1.1.m1.1b" encoding="MathML-Content">
+ <apply xref="p1.1.m1.1.3" id="p1.1.m1.1.3.cmml">
+ <csymbol id="p1.1.m1.1.3.1.cmml" cd="ambiguous">subscript</csymbol>
+ <ci xref="p1.1.m1.1.1" id="p1.1.m1.1.1.cmml">ðµ</ci>
+ <list xref="p1.1.m1.1.2.1" id="p1.1.m1.1.2.1.4.cmml">
+ <ci xref="p1.1.m1.1.2.1.1" id="p1.1.m1.1.2.1.1.cmml">ð‘›</ci>
+ <ci xref="p1.1.m1.1.2.1.3" id="p1.1.m1.1.2.1.3.cmml">ð‘š</ci>
+ </list>
+ </apply>
+ </annotation-xml>
+ <annotation id="p1.1.m1.1c" encoding="application/x-tex">B_{{n,m}}</annotation>
+ </semantics>
+</math>, cf. <code class="func">IsIntegralCyclotomic</code> (<a shape="rect" href="chap18_mj.html#X869750DA81EA0E67"><span class="RefLink">18.1-4</span></a>)), it is a normal basis for squarefree <math id="5805483341809995655" display="inline" alttext="n" class="ltx_Math" xmlns="http://www.w3.org/1998/Math/MathML">
+ <semantics id="p1.1.m1.1a">
+ <mi xref="p1.1.m1.1.1.cmml" id="p1.1.m1.1.1">n</mi>
+ <annotation-xml id="p1.1.m1.1b" encoding="MathML-Content">
+ <ci xref="p1.1.m1.1.1" id="p1.1.m1.1.1.cmml">ð‘›</ci>
+ </annotation-xml>
+ <annotation id="p1.1.m1.1c" encoding="application/x-tex">n</annotation>
+ </semantics>
+</math> and closed under complex conjugation for odd <math id="415159508013058527" display="inline" alttext="n" class="ltx_Math" xmlns="http://www.w3.org/1998/Math/MathML">
+ <semantics id="p1.1.m1.1a">
+ <mi xref="p1.1.m1.1.1.cmml" id="p1.1.m1.1.1">n</mi>
+ <annotation-xml id="p1.1.m1.1b" encoding="MathML-Content">
+ <ci xref="p1.1.m1.1.1" id="p1.1.m1.1.1.cmml">ð‘›</ci>
+ </annotation-xml>
+ <annotation id="p1.1.m1.1c" encoding="application/x-tex">n</annotation>
+ </semantics>
+</math>.</p><p xmlns="http://www.w3.org/1999/xhtml"><em>Note:</em> For <math id="2208151708259454343" display="inline" alttext="\textit{n}\equiv 2\pmod{4}" class="ltx_Math" xmlns="http://www.w3.org/1998/Math/MathML">
+ <semantics id="p1.1.m1.1a">
+ <mrow xref="p1.1.m1.1.5.cmml" id="p1.1.m1.1.5">
+ <mtext xref="p1.1.m1.1.1.1.cmml" id="p1.1.m1.1.1.1">ð‘›</mtext>
+ <mo xref="p1.1.m1.1.2.cmml" id="p1.1.m1.1.2">≡</mo>
+ <mrow xref="p1.1.m1.1.5.1.cmml" id="p1.1.m1.1.5.1">
+ <mn xref="p1.1.m1.1.3.cmml" id="p1.1.m1.1.3">2</mn>
+ <mspace xref="p1.1.m1.1.5.1.cmml" id="p1.1.m1.1.5.1a" width="veryverythickmathspace"/>
+ <mrow xref="p1.1.m1.1.4.1.cmml" id="p1.1.m1.1.4.10">
+ <mo xref="p1.1.m1.1.4.1.1.cmml" id="p1.1.m1.1.4.4" stretchy="false" lspace="8.1pt">(</mo>
+ <mrow xref="p1.1.m1.1.4.1.cmml" id="p1.1.m1.1.4.10.2">
+ <mo xref="p1.1.m1.1.4.1.1.cmml" id="p1.1.m1.1.4.7">mod</mo>
+ <mn xref="p1.1.m1.1.4.8.cmml" id="p1.1.m1.1.4.8">4</mn>
+ </mrow>
+ <mo xref="p1.1.m1.1.4.1.1.cmml" id="p1.1.m1.1.4.9" stretchy="false">)</mo>
+ </mrow>
+ </mrow>
+ </mrow>
+ <annotation-xml id="p1.1.m1.1b" encoding="MathML-Content">
+ <apply xref="p1.1.m1.1.5" id="p1.1.m1.1.5.cmml">
+ <equivalent xref="p1.1.m1.1.2" id="p1.1.m1.1.2.cmml"/>
+ <mtext xref="p1.1.m1.1.1.1" id="p1.1.m1.1.1.1.cmml">
+ <span class="ltx_text ltx_font_italic" xmlns="http://www.w3.org/1999/xhtml">n</span>
+ </mtext>
+ <apply xref="p1.1.m1.1.5.1" id="p1.1.m1.1.5.1.cmml">
+ <csymbol id="p1.1.m1.1.5.1.1.cmml" cd="latexml">annotated</csymbol>
+ <cn xref="p1.1.m1.1.3" id="p1.1.m1.1.3.cmml" type="integer">2</cn>
+ <apply xref="p1.1.m1.1.4.10" id="p1.1.m1.1.4.1.cmml">
+ <ci xref="p1.1.m1.1.4.4" id="p1.1.m1.1.4.1.1.cmml">pmod</ci>
+ <cn xref="p1.1.m1.1.4.8" id="p1.1.m1.1.4.8.cmml" type="integer">4</cn>
+ </apply>
+ </apply>
+ </apply>
+ </annotation-xml>
+ <annotation id="p1.1.m1.1c" encoding="application/x-tex">\textit{n}\equiv 2\pmod{4}</annotation>
+ </semantics>
+</math>, we have <code class="code">ZumbroichBase(<var class="Arg">n</var>, 1) = 2 * ZumbroichBase(<var class="Arg">n</var>/2, 1)</code> and <code class="code">List( ZumbroichBase(<var class="Arg">n</var>, 1), x -> E(<var class="Arg">n</var>)^x ) = List( ZumbroichBase(<var class="Arg">n</var>/2, 1), x -> E(<var class="Arg">n</var>/2)^x )</code>.</p><div class="example" xmlns="http://www.w3.org/1999/xhtml"><pre xml:space="preserve">
+<span class="GAPprompt">gap></span> <span class="GAPinput">ZumbroichBase( 15, 1 ); ZumbroichBase( 12, 3 );</span>
+[ 1, 2, 4, 7, 8, 11, 13, 14 ]
+[ 0, 3 ]
+<span class="GAPprompt">gap></span> <span class="GAPinput">ZumbroichBase( 10, 2 ); ZumbroichBase( 32, 4 );</span>
+[ 2, 4, 6, 8 ]
+[ 0, 1, 2, 3, 4, 5, 6, 7 ]
+</pre></div> </body>
+ </html>
\ No newline at end of file
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new file mode 100644
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--- /dev/null
+++ b/export/mwscrawler/15.html
@@ -0,0 +1,202 @@
+<html>
+ <head>
+ <title>60.3 Integral Bases of Abelian Number Fields</title>
+ <meta name="url" content="http://www.gap-system.org/Manuals/doc/ref/TODO#X7D2421AC8491D2BE"></meta>
+ </head>
+ <body> <h4 xmlns="http://www.w3.org/1999/xhtml">60.3 <span class="Heading">Integral Bases of Abelian Number Fields</span></h4><p xmlns="http://www.w3.org/1999/xhtml">Each abelian number field is naturally a vector space over <math id="-4243951106085483006" display="inline" alttext="?" class="ltx_Math" xmlns="http://www.w3.org/1998/Math/MathML">
+ <semantics id="p1.1.m1.1a">
+ <mi xref="p1.1.m1.1.1.cmml" id="p1.1.m1.1.1" mathvariant="normal">?</mi>
+ <annotation-xml id="p1.1.m1.1b" encoding="MathML-Content">
+ <ci xref="p1.1.m1.1.1" id="p1.1.m1.1.1.cmml">?</ci>
+ </annotation-xml>
+ <annotation id="p1.1.m1.1c" encoding="application/x-tex">?</annotation>
+ </semantics>
+</math>. Moreover, if the abelian number field <math id="-8908099324234125357" display="inline" alttext="F" class="ltx_Math" xmlns="http://www.w3.org/1998/Math/MathML">
+ <semantics id="p1.1.m1.1a">
+ <mi xref="p1.1.m1.1.1.cmml" id="p1.1.m1.1.1">F</mi>
+ <annotation-xml id="p1.1.m1.1b" encoding="MathML-Content">
+ <ci xref="p1.1.m1.1.1" id="p1.1.m1.1.1.cmml">ð¹</ci>
+ </annotation-xml>
+ <annotation id="p1.1.m1.1c" encoding="application/x-tex">F</annotation>
+ </semantics>
+</math> contains the <math id="3138850455013136997" display="inline" alttext="n" class="ltx_Math" xmlns="http://www.w3.org/1998/Math/MathML">
+ <semantics id="p1.1.m1.1a">
+ <mi xref="p1.1.m1.1.1.cmml" id="p1.1.m1.1.1">n</mi>
+ <annotation-xml id="p1.1.m1.1b" encoding="MathML-Content">
+ <ci xref="p1.1.m1.1.1" id="p1.1.m1.1.1.cmml">ð‘›</ci>
+ </annotation-xml>
+ <annotation id="p1.1.m1.1c" encoding="application/x-tex">n</annotation>
+ </semantics>
+</math>-th cyclotomic field <math id="-4090466234368629443" display="inline" alttext="?_{n}" class="ltx_Math" xmlns="http://www.w3.org/1998/Math/MathML">
+ <semantics id="p1.1.m1.1a">
+ <msub xref="p1.1.m1.1.3.cmml" id="p1.1.m1.1.3">
+ <mi xref="p1.1.m1.1.1.cmml" id="p1.1.m1.1.1" mathvariant="normal">?</mi>
+ <mi xref="p1.1.m1.1.2.1.cmml" id="p1.1.m1.1.2.1">n</mi>
+ </msub>
+ <annotation-xml id="p1.1.m1.1b" encoding="MathML-Content">
+ <apply xref="p1.1.m1.1.3" id="p1.1.m1.1.3.cmml">
+ <csymbol id="p1.1.m1.1.3.1.cmml" cd="ambiguous">subscript</csymbol>
+ <ci xref="p1.1.m1.1.1" id="p1.1.m1.1.1.cmml">?</ci>
+ <ci xref="p1.1.m1.1.2.1" id="p1.1.m1.1.2.1.cmml">ð‘›</ci>
+ </apply>
+ </annotation-xml>
+ <annotation id="p1.1.m1.1c" encoding="application/x-tex">?_{n}</annotation>
+ </semantics>
+</math> then <math id="-421311888980028774" display="inline" alttext="F" class="ltx_Math" xmlns="http://www.w3.org/1998/Math/MathML">
+ <semantics id="p1.1.m1.1a">
+ <mi xref="p1.1.m1.1.1.cmml" id="p1.1.m1.1.1">F</mi>
+ <annotation-xml id="p1.1.m1.1b" encoding="MathML-Content">
+ <ci xref="p1.1.m1.1.1" id="p1.1.m1.1.1.cmml">ð¹</ci>
+ </annotation-xml>
+ <annotation id="p1.1.m1.1c" encoding="application/x-tex">F</annotation>
+ </semantics>
+</math> is a vector space over <math id="7414210350085120196" display="inline" alttext="?_{n}" class="ltx_Math" xmlns="http://www.w3.org/1998/Math/MathML">
+ <semantics id="p1.1.m1.1a">
+ <msub xref="p1.1.m1.1.3.cmml" id="p1.1.m1.1.3">
+ <mi xref="p1.1.m1.1.1.cmml" id="p1.1.m1.1.1" mathvariant="normal">?</mi>
+ <mi xref="p1.1.m1.1.2.1.cmml" id="p1.1.m1.1.2.1">n</mi>
+ </msub>
+ <annotation-xml id="p1.1.m1.1b" encoding="MathML-Content">
+ <apply xref="p1.1.m1.1.3" id="p1.1.m1.1.3.cmml">
+ <csymbol id="p1.1.m1.1.3.1.cmml" cd="ambiguous">subscript</csymbol>
+ <ci xref="p1.1.m1.1.1" id="p1.1.m1.1.1.cmml">?</ci>
+ <ci xref="p1.1.m1.1.2.1" id="p1.1.m1.1.2.1.cmml">ð‘›</ci>
+ </apply>
+ </annotation-xml>
+ <annotation id="p1.1.m1.1c" encoding="application/x-tex">?_{n}</annotation>
+ </semantics>
+</math>. In <strong class="pkg">GAP</strong>, each field object represents a vector space object over a certain subfield <math id="1801983466100099123" display="inline" alttext="S" class="ltx_Math" xmlns="http://www.w3.org/1998/Math/MathML">
+ <semantics id="p1.1.m1.1a">
+ <mi xref="p1.1.m1.1.1.cmml" id="p1.1.m1.1.1">S</mi>
+ <annotation-xml id="p1.1.m1.1b" encoding="MathML-Content">
+ <ci xref="p1.1.m1.1.1" id="p1.1.m1.1.1.cmml">ð‘†</ci>
+ </annotation-xml>
+ <annotation id="p1.1.m1.1c" encoding="application/x-tex">S</annotation>
+ </semantics>
+</math>, which depends on the way <math id="-1517757394707115667" display="inline" alttext="F" class="ltx_Math" xmlns="http://www.w3.org/1998/Math/MathML">
+ <semantics id="p1.1.m1.1a">
+ <mi xref="p1.1.m1.1.1.cmml" id="p1.1.m1.1.1">F</mi>
+ <annotation-xml id="p1.1.m1.1b" encoding="MathML-Content">
+ <ci xref="p1.1.m1.1.1" id="p1.1.m1.1.1.cmml">ð¹</ci>
+ </annotation-xml>
+ <annotation id="p1.1.m1.1c" encoding="application/x-tex">F</annotation>
+ </semantics>
+</math> was constructed. The subfield <math id="7045353852483731115" display="inline" alttext="S" class="ltx_Math" xmlns="http://www.w3.org/1998/Math/MathML">
+ <semantics id="p1.1.m1.1a">
+ <mi xref="p1.1.m1.1.1.cmml" id="p1.1.m1.1.1">S</mi>
+ <annotation-xml id="p1.1.m1.1b" encoding="MathML-Content">
+ <ci xref="p1.1.m1.1.1" id="p1.1.m1.1.1.cmml">ð‘†</ci>
+ </annotation-xml>
+ <annotation id="p1.1.m1.1c" encoding="application/x-tex">S</annotation>
+ </semantics>
+</math> can be accessed as the value of the attribute <code class="func">LeftActingDomain</code> (<a shape="rect" href="chap57_mj.html#X86F070E0807DC34E"><span class="RefLink">57.1-11</span></a>).</p><p xmlns="http://www.w3.org/1999/xhtml">The return values of <code class="func">NF</code> (<a shape="rect" href="chap60_mj.html#X80E5AD028143E11E"><span class="RefLink">60.1-2</span></a>) and of the one argument versions of <code class="func">CF</code> (<a shape="rect" href="chap60_mj.html#X80D21D80850EFA4B"><span class="RefLink">60.1-1</span></a>) represent vector spaces over <math id="-964803556485730101" display="inline" alttext="?" class="ltx_Math" xmlns="http://www.w3.org/1998/Math/MathML">
+ <semantics id="p1.1.m1.1a">
+ <mi xref="p1.1.m1.1.1.cmml" id="p1.1.m1.1.1" mathvariant="normal">?</mi>
+ <annotation-xml id="p1.1.m1.1b" encoding="MathML-Content">
+ <ci xref="p1.1.m1.1.1" id="p1.1.m1.1.1.cmml">?</ci>
+ </annotation-xml>
+ <annotation id="p1.1.m1.1c" encoding="application/x-tex">?</annotation>
+ </semantics>
+</math>, and the return values of the two argument version of <code class="func">CF</code> (<a shape="rect" href="chap60_mj.html#X80D21D80850EFA4B"><span class="RefLink">60.1-1</span></a>) represent vector spaces over the field that is given as the first argument. For an abelian number field <var class="Arg">F</var> and a subfield <var class="Arg">S</var> of <var class="Arg">F</var>, a <strong class="pkg">GAP</strong> object representing <var class="Arg">F</var> as a vector space over <var class="Arg">S</var> can be constructed using <code class="func">AsField</code> (<a shape="rect" href="chap58_mj.html#X7C193B7D7AFB29BE"><span class="RefLink">58.1-9</span></a>).</p><p xmlns="http://www.w3.org/1999/xhtml">Let <var class="Arg">F</var> be the cyclotomic field <math id="995609940989272001" display="inline" alttext="?_{n}" class="ltx_Math" xmlns="http://www.w3.org/1998/Math/MathML">
+ <semantics id="p1.1.m1.1a">
+ <msub xref="p1.1.m1.1.3.cmml" id="p1.1.m1.1.3">
+ <mi xref="p1.1.m1.1.1.cmml" id="p1.1.m1.1.1" mathvariant="normal">?</mi>
+ <mi xref="p1.1.m1.1.2.1.cmml" id="p1.1.m1.1.2.1">n</mi>
+ </msub>
+ <annotation-xml id="p1.1.m1.1b" encoding="MathML-Content">
+ <apply xref="p1.1.m1.1.3" id="p1.1.m1.1.3.cmml">
+ <csymbol id="p1.1.m1.1.3.1.cmml" cd="ambiguous">subscript</csymbol>
+ <ci xref="p1.1.m1.1.1" id="p1.1.m1.1.1.cmml">?</ci>
+ <ci xref="p1.1.m1.1.2.1" id="p1.1.m1.1.2.1.cmml">ð‘›</ci>
+ </apply>
+ </annotation-xml>
+ <annotation id="p1.1.m1.1c" encoding="application/x-tex">?_{n}</annotation>
+ </semantics>
+</math>, represented as a vector space over the subfield <var class="Arg">S</var>. If <var class="Arg">S</var> is the cyclotomic field <math id="2376284400051401582" display="inline" alttext="?_{m}" class="ltx_Math" xmlns="http://www.w3.org/1998/Math/MathML">
+ <semantics id="p1.1.m1.1a">
+ <msub xref="p1.1.m1.1.3.cmml" id="p1.1.m1.1.3">
+ <mi xref="p1.1.m1.1.1.cmml" id="p1.1.m1.1.1" mathvariant="normal">?</mi>
+ <mi xref="p1.1.m1.1.2.1.cmml" id="p1.1.m1.1.2.1">m</mi>
+ </msub>
+ <annotation-xml id="p1.1.m1.1b" encoding="MathML-Content">
+ <apply xref="p1.1.m1.1.3" id="p1.1.m1.1.3.cmml">
+ <csymbol id="p1.1.m1.1.3.1.cmml" cd="ambiguous">subscript</csymbol>
+ <ci xref="p1.1.m1.1.1" id="p1.1.m1.1.1.cmml">?</ci>
+ <ci xref="p1.1.m1.1.2.1" id="p1.1.m1.1.2.1.cmml">ð‘š</ci>
+ </apply>
+ </annotation-xml>
+ <annotation id="p1.1.m1.1c" encoding="application/x-tex">?_{m}</annotation>
+ </semantics>
+</math>, with <math id="-8433364101409720809" display="inline" alttext="m" class="ltx_Math" xmlns="http://www.w3.org/1998/Math/MathML">
+ <semantics id="p1.1.m1.1a">
+ <mi xref="p1.1.m1.1.1.cmml" id="p1.1.m1.1.1">m</mi>
+ <annotation-xml id="p1.1.m1.1b" encoding="MathML-Content">
+ <ci xref="p1.1.m1.1.1" id="p1.1.m1.1.1.cmml">ð‘š</ci>
+ </annotation-xml>
+ <annotation id="p1.1.m1.1c" encoding="application/x-tex">m</annotation>
+ </semantics>
+</math> a divisor of <math id="627653283343250934" display="inline" alttext="n" class="ltx_Math" xmlns="http://www.w3.org/1998/Math/MathML">
+ <semantics id="p1.1.m1.1a">
+ <mi xref="p1.1.m1.1.1.cmml" id="p1.1.m1.1.1">n</mi>
+ <annotation-xml id="p1.1.m1.1b" encoding="MathML-Content">
+ <ci xref="p1.1.m1.1.1" id="p1.1.m1.1.1.cmml">ð‘›</ci>
+ </annotation-xml>
+ <annotation id="p1.1.m1.1c" encoding="application/x-tex">n</annotation>
+ </semantics>
+</math>, then <code class="code">CanonicalBasis( <var class="Arg">F</var> )</code> returns the Zumbroich basis of <var class="Arg">F</var> relative to <var class="Arg">S</var>, which consists of the roots of unity <code class="code">E(<var class="Arg">n</var>)</code>^<var class="Arg">i</var> where <var class="Arg">i</var> is an element of the list <code class="code">ZumbroichBase( <var class="Arg">n</var>, <var class="Arg">m</var> )</code> (see <code class="func">ZumbroichBase</code> (<a shape="rect" href="chap60_mj.html#X7F52BEA0862E06F2"><span class="RefLink">60.3-1</span></a>)). If <var class="Arg">S</var> is an abelian number field that is not a cyclotomic field then <code class="code">CanonicalBasis( <var class="Arg">F</var> )</code> returns a normal <var class="Arg">S</var>-basis of <var class="Arg">F</var>, i.e., a basis that is closed under the field automorphisms of <var class="Arg">F</var>.</p><p xmlns="http://www.w3.org/1999/xhtml">Let <var class="Arg">F</var> be the abelian number field <code class="code">NF( <var class="Arg">n</var>, <var class="Arg">stab</var> )</code>, with conductor <var class="Arg">n</var>, that is itself not a cyclotomic field, represented as a vector space over the subfield <var class="Arg">S</var>. If <var class="Arg">S</var> is the cyclotomic field <math id="-576215802115201341" display="inline" alttext="?_{m}" class="ltx_Math" xmlns="http://www.w3.org/1998/Math/MathML">
+ <semantics id="p1.1.m1.1a">
+ <msub xref="p1.1.m1.1.3.cmml" id="p1.1.m1.1.3">
+ <mi xref="p1.1.m1.1.1.cmml" id="p1.1.m1.1.1" mathvariant="normal">?</mi>
+ <mi xref="p1.1.m1.1.2.1.cmml" id="p1.1.m1.1.2.1">m</mi>
+ </msub>
+ <annotation-xml id="p1.1.m1.1b" encoding="MathML-Content">
+ <apply xref="p1.1.m1.1.3" id="p1.1.m1.1.3.cmml">
+ <csymbol id="p1.1.m1.1.3.1.cmml" cd="ambiguous">subscript</csymbol>
+ <ci xref="p1.1.m1.1.1" id="p1.1.m1.1.1.cmml">?</ci>
+ <ci xref="p1.1.m1.1.2.1" id="p1.1.m1.1.2.1.cmml">ð‘š</ci>
+ </apply>
+ </annotation-xml>
+ <annotation id="p1.1.m1.1c" encoding="application/x-tex">?_{m}</annotation>
+ </semantics>
+</math>, with <math id="-8773796078956449250" display="inline" alttext="m" class="ltx_Math" xmlns="http://www.w3.org/1998/Math/MathML">
+ <semantics id="p1.1.m1.1a">
+ <mi xref="p1.1.m1.1.1.cmml" id="p1.1.m1.1.1">m</mi>
+ <annotation-xml id="p1.1.m1.1b" encoding="MathML-Content">
+ <ci xref="p1.1.m1.1.1" id="p1.1.m1.1.1.cmml">ð‘š</ci>
+ </annotation-xml>
+ <annotation id="p1.1.m1.1c" encoding="application/x-tex">m</annotation>
+ </semantics>
+</math> a divisor of <math id="-5692834966176752574" display="inline" alttext="n" class="ltx_Math" xmlns="http://www.w3.org/1998/Math/MathML">
+ <semantics id="p1.1.m1.1a">
+ <mi xref="p1.1.m1.1.1.cmml" id="p1.1.m1.1.1">n</mi>
+ <annotation-xml id="p1.1.m1.1b" encoding="MathML-Content">
+ <ci xref="p1.1.m1.1.1" id="p1.1.m1.1.1.cmml">ð‘›</ci>
+ </annotation-xml>
+ <annotation id="p1.1.m1.1c" encoding="application/x-tex">n</annotation>
+ </semantics>
+</math>, then <code class="code">CanonicalBasis( <var class="Arg">F</var> )</code> returns the Lenstra basis of <var class="Arg">F</var> relative to <var class="Arg">S</var> that consists of the sums of roots of unity described by <code class="code">LenstraBase( <var class="Arg">n</var>, <var class="Arg">stab</var>, <var class="Arg">stab</var>, <var class="Arg">m</var> )</code> (see <code class="func">LenstraBase</code> (<a shape="rect" href="chap60_mj.html#X87DB9C2C858B722A"><span class="RefLink">60.3-2</span></a>)). If <var class="Arg">S</var> is an abelian number field that is not a cyclotomic field then <code class="code">CanonicalBasis( <var class="Arg">F</var> )</code> returns a normal <var class="Arg">S</var>-basis of <var class="Arg">F</var>.</p><div class="example" xmlns="http://www.w3.org/1999/xhtml"><pre xml:space="preserve">
+<span class="GAPprompt">gap></span> <span class="GAPinput">f:= CF(8);; # a cycl. field over the rationals</span>
+<span class="GAPprompt">gap></span> <span class="GAPinput">b:= CanonicalBasis( f );; BasisVectors( b );</span>
+[ 1, E(8), E(4), E(8)^3 ]
+<span class="GAPprompt">gap></span> <span class="GAPinput">Coefficients( b, Sqrt(-2) );</span>
+[ 0, 1, 0, 1 ]
+<span class="GAPprompt">gap></span> <span class="GAPinput">f:= AsField( CF(4), CF(8) );; # a cycl. field over a cycl. field</span>
+<span class="GAPprompt">gap></span> <span class="GAPinput">b:= CanonicalBasis( f );; BasisVectors( b );</span>
+[ 1, E(8) ]
+<span class="GAPprompt">gap></span> <span class="GAPinput">Coefficients( b, Sqrt(-2) );</span>
+[ 0, 1+E(4) ]
+<span class="GAPprompt">gap></span> <span class="GAPinput">f:= AsField( Field( [ Sqrt(-2) ] ), CF(8) );;</span>
+<span class="GAPprompt">gap></span> <span class="GAPinput"># a cycl. field over a non-cycl. field</span>
+<span class="GAPprompt">gap></span> <span class="GAPinput">b:= CanonicalBasis( f );; BasisVectors( b );</span>
+[ 1/2+1/2*E(8)-1/2*E(8)^2-1/2*E(8)^3,
+ 1/2-1/2*E(8)+1/2*E(8)^2+1/2*E(8)^3 ]
+<span class="GAPprompt">gap></span> <span class="GAPinput">Coefficients( b, Sqrt(-2) );</span>
+[ E(8)+E(8)^3, E(8)+E(8)^3 ]
+<span class="GAPprompt">gap></span> <span class="GAPinput">f:= Field( [ Sqrt(-2) ] ); # a non-cycl. field over the rationals</span>
+NF(8,[ 1, 3 ])
+<span class="GAPprompt">gap></span> <span class="GAPinput">b:= CanonicalBasis( f );; BasisVectors( b );</span>
+[ 1, E(8)+E(8)^3 ]
+<span class="GAPprompt">gap></span> <span class="GAPinput">Coefficients( b, Sqrt(-2) );</span>
+[ 0, 1 ]
+</pre></div> </body>
+ </html>
\ No newline at end of file
diff --git a/export/mwscrawler/16.html b/export/mwscrawler/16.html
new file mode 100644
index 0000000..8f38557
--- /dev/null
+++ b/export/mwscrawler/16.html
@@ -0,0 +1,78 @@
+<html>
+ <head>
+ <title>60.2-5 GaloisStabilizer</title>
+ <meta name="url" content="http://www.gap-system.org/Manuals/doc/ref/TODO#X87E7313D8070B9CC"></meta>
+ </head>
+ <body> <h5 xmlns="http://www.w3.org/1999/xhtml">60.2-5 GaloisStabilizer</h5><div class="func" xmlns="http://www.w3.org/1999/xhtml"><table width="100%" class="func"><tbody><tr><td colspan="1" rowspan="1" class="tdleft"><code class="func">‣ GaloisStabilizer</code>( <var class="Arg">F</var> )</td><td colspan="1" rowspan="1" class="tdright">( attribute )</td></tr></tbody></table></div><p xmlns="http://www.w3.org/1999/xhtml">Let <var class="Arg">F</var> be an abelian number field (see <code class="func">IsAbelianNumberField</code> (<a shape="rect" href="chap60_mj.html#X7D202D707D5708FA"><span class="RefLink">60.2-3</span></a>)) with conductor <math id="-4737294481319410573" display="inline" alttext="n" class="ltx_Math" xmlns="http://www.w3.org/1998/Math/MathML">
+ <semantics id="p1.1.m1.1a">
+ <mi xref="p1.1.m1.1.1.cmml" id="p1.1.m1.1.1">n</mi>
+ <annotation-xml id="p1.1.m1.1b" encoding="MathML-Content">
+ <ci xref="p1.1.m1.1.1" id="p1.1.m1.1.1.cmml">ð‘›</ci>
+ </annotation-xml>
+ <annotation id="p1.1.m1.1c" encoding="application/x-tex">n</annotation>
+ </semantics>
+</math>, say. (This means that the <math id="-1251700905668443816" display="inline" alttext="n" class="ltx_Math" xmlns="http://www.w3.org/1998/Math/MathML">
+ <semantics id="p1.1.m1.1a">
+ <mi xref="p1.1.m1.1.1.cmml" id="p1.1.m1.1.1">n</mi>
+ <annotation-xml id="p1.1.m1.1b" encoding="MathML-Content">
+ <ci xref="p1.1.m1.1.1" id="p1.1.m1.1.1.cmml">ð‘›</ci>
+ </annotation-xml>
+ <annotation id="p1.1.m1.1c" encoding="application/x-tex">n</annotation>
+ </semantics>
+</math>-th cyclotomic field is the smallest cyclotomic field containing <var class="Arg">F</var>, see <code class="func">Conductor</code> (<a shape="rect" href="chap18_mj.html#X815D6EC57CBA9827"><span class="RefLink">18.1-7</span></a>).) <code class="func">GaloisStabilizer</code> returns the set of all those integers <math id="-7493539776088871027" display="inline" alttext="k" class="ltx_Math" xmlns="http://www.w3.org/1998/Math/MathML">
+ <semantics id="p1.1.m1.1a">
+ <mi xref="p1.1.m1.1.1.cmml" id="p1.1.m1.1.1">k</mi>
+ <annotation-xml id="p1.1.m1.1b" encoding="MathML-Content">
+ <ci xref="p1.1.m1.1.1" id="p1.1.m1.1.1.cmml">ð‘˜</ci>
+ </annotation-xml>
+ <annotation id="p1.1.m1.1c" encoding="application/x-tex">k</annotation>
+ </semantics>
+</math> in the range <math id="-2959061408037581052" display="inline" alttext="[1..n]" class="ltx_Math" xmlns="http://www.w3.org/1998/Math/MathML">
+ <semantics id="p1.1.m1.1a">
+ <mrow id="p1.1.m1.1b">
+ <mo xref="p1.1.m1.1.1.cmml" id="p1.1.m1.1.1" stretchy="false">[</mo>
+ <mn xref="p1.1.m1.1.2.cmml" id="p1.1.m1.1.2">1</mn>
+ <mo xref="p1.1.m1.1.3.cmml" id="p1.1.m1.1.3">.</mo>
+ <mo xref="p1.1.m1.1.4.cmml" id="p1.1.m1.1.4">.</mo>
+ <mi id="p1.1.m1.1.5">n</mi>
+ <mo xref="p1.1.m1.1.6.cmml" id="p1.1.m1.1.6" stretchy="false">]</mo>
+ </mrow>
+ <annotation-xml id="p1.1.m1.1c" encoding="MathML-Content">
+ <cerror id="p1.1.m1.1d">
+ <csymbol id="p1.1.m1.1e" cd="ambiguous">fragments</csymbol>
+ <ci xref="p1.1.m1.1.1" id="p1.1.m1.1.1.cmml">[</ci>
+ <cn xref="p1.1.m1.1.2" id="p1.1.m1.1.2.cmml" type="integer">1</cn>
+ <ci xref="p1.1.m1.1.3" id="p1.1.m1.1.3.cmml">.</ci>
+ <ci xref="p1.1.m1.1.4" id="p1.1.m1.1.4.cmml">.</ci>
+ <csymbol id="p1.1.m1.1f" cd="unknown">n</csymbol>
+ <ci xref="p1.1.m1.1.6" id="p1.1.m1.1.6.cmml">]</ci>
+ </cerror>
+ </annotation-xml>
+ <annotation id="p1.1.m1.1g" encoding="application/x-tex">[1..n]</annotation>
+ </semantics>
+</math> such that the field automorphism induced by raising <math id="-6979597911055838736" display="inline" alttext="n" class="ltx_Math" xmlns="http://www.w3.org/1998/Math/MathML">
+ <semantics id="p1.1.m1.1a">
+ <mi xref="p1.1.m1.1.1.cmml" id="p1.1.m1.1.1">n</mi>
+ <annotation-xml id="p1.1.m1.1b" encoding="MathML-Content">
+ <ci xref="p1.1.m1.1.1" id="p1.1.m1.1.1.cmml">ð‘›</ci>
+ </annotation-xml>
+ <annotation id="p1.1.m1.1c" encoding="application/x-tex">n</annotation>
+ </semantics>
+</math>-th roots of unity to the <math id="-4443708858965436565" display="inline" alttext="k" class="ltx_Math" xmlns="http://www.w3.org/1998/Math/MathML">
+ <semantics id="p1.1.m1.1a">
+ <mi xref="p1.1.m1.1.1.cmml" id="p1.1.m1.1.1">k</mi>
+ <annotation-xml id="p1.1.m1.1b" encoding="MathML-Content">
+ <ci xref="p1.1.m1.1.1" id="p1.1.m1.1.1.cmml">ð‘˜</ci>
+ </annotation-xml>
+ <annotation id="p1.1.m1.1c" encoding="application/x-tex">k</annotation>
+ </semantics>
+</math>-th power acts trivially on <var class="Arg">F</var>.</p><div class="example" xmlns="http://www.w3.org/1999/xhtml"><pre xml:space="preserve">
+<span class="GAPprompt">gap></span> <span class="GAPinput">r5:= Sqrt(5);</span>
+E(5)-E(5)^2-E(5)^3+E(5)^4
+<span class="GAPprompt">gap></span> <span class="GAPinput">GaloisCyc( r5, 4 ) = r5; GaloisCyc( r5, 2 ) = r5;</span>
+true
+false
+<span class="GAPprompt">gap></span> <span class="GAPinput">GaloisStabilizer( Field( [ r5 ] ) );</span>
+[ 1, 4 ]
+</pre></div> </body>
+ </html>
\ No newline at end of file
diff --git a/export/mwscrawler/17.html b/export/mwscrawler/17.html
new file mode 100644
index 0000000..7e25ab5
--- /dev/null
+++ b/export/mwscrawler/17.html
@@ -0,0 +1,19 @@
+<html>
+ <head>
+ <title>60.2-4 IsCyclotomicField</title>
+ <meta name="url" content="http://www.gap-system.org/Manuals/doc/ref/TODO#X84CAE4627F0CD639"></meta>
+ </head>
+ <body> <h5 xmlns="http://www.w3.org/1999/xhtml">60.2-4 IsCyclotomicField</h5><div class="func" xmlns="http://www.w3.org/1999/xhtml"><table width="100%" class="func"><tbody><tr><td colspan="1" rowspan="1" class="tdleft"><code class="func">‣ IsCyclotomicField</code>( <var class="Arg">F</var> )</td><td colspan="1" rowspan="1" class="tdright">( property )</td></tr></tbody></table></div><p xmlns="http://www.w3.org/1999/xhtml">returns <code class="keyw">true</code> if the field <var class="Arg">F</var> is a <em>cyclotomic field</em>, i.e., an abelian number field (see <code class="func">IsAbelianNumberField</code> (<a shape="rect" href="chap60_mj.html#X7D202D707D5708FA"><span class="RefLink">60.2-3</span></a>)) that can be generated by roots of unity.</p><div class="example" xmlns="http://www.w3.org/1999/xhtml"><pre xml:space="preserve">
+<span class="GAPprompt">gap></span> <span class="GAPinput">IsNumberField( CF(9) ); IsAbelianNumberField( Field( [ ER(3) ] ) );</span>
+true
+true
+<span class="GAPprompt">gap></span> <span class="GAPinput">IsNumberField( GF(2) );</span>
+false
+<span class="GAPprompt">gap></span> <span class="GAPinput">IsCyclotomicField( CF(9) );</span>
+true
+<span class="GAPprompt">gap></span> <span class="GAPinput">IsCyclotomicField( Field( [ Sqrt(-3) ] ) );</span>
+true
+<span class="GAPprompt">gap></span> <span class="GAPinput">IsCyclotomicField( Field( [ Sqrt(3) ] ) );</span>
+false
+</pre></div> </body>
+ </html>
\ No newline at end of file
diff --git a/export/mwscrawler/18.html b/export/mwscrawler/18.html
new file mode 100644
index 0000000..b9654bf
--- /dev/null
+++ b/export/mwscrawler/18.html
@@ -0,0 +1,7 @@
+<html>
+ <head>
+ <title>60.2-3 IsAbelianNumberField</title>
+ <meta name="url" content="http://www.gap-system.org/Manuals/doc/ref/TODO#X7D202D707D5708FA"></meta>
+ </head>
+ <body> <h5 xmlns="http://www.w3.org/1999/xhtml">60.2-3 IsAbelianNumberField</h5><div class="func" xmlns="http://www.w3.org/1999/xhtml"><table width="100%" class="func"><tbody><tr><td colspan="1" rowspan="1" class="tdleft"><code class="func">‣ IsAbelianNumberField</code>( <var class="Arg">F</var> )</td><td colspan="1" rowspan="1" class="tdright">( property )</td></tr></tbody></table></div><p xmlns="http://www.w3.org/1999/xhtml">returns <code class="keyw">true</code> if the field <var class="Arg">F</var> is a number field (see <code class="func">IsNumberField</code> (<a shape="rect" href="chap60_mj.html#X87D78F5E875F2E8A"><span class="RefLink">60.2-2</span></a>)) that is a Galois extension of the prime field, with abelian Galois group (see <code class="func">GaloisGroup</code> (<a shape="rect" href="chap58_mj.html#X80CAA5BA82F09ED2"><span class="RefLink">58.3-1</span></a>)).</p> </body>
+ </html>
\ No newline at end of file
diff --git a/export/mwscrawler/19.html b/export/mwscrawler/19.html
new file mode 100644
index 0000000..c8918b5
--- /dev/null
+++ b/export/mwscrawler/19.html
@@ -0,0 +1,7 @@
+<html>
+ <head>
+ <title>60.2-2 IsNumberField</title>
+ <meta name="url" content="http://www.gap-system.org/Manuals/doc/ref/TODO#X87D78F5E875F2E8A"></meta>
+ </head>
+ <body> <h5 xmlns="http://www.w3.org/1999/xhtml">60.2-2 IsNumberField</h5><div class="func" xmlns="http://www.w3.org/1999/xhtml"><table width="100%" class="func"><tbody><tr><td colspan="1" rowspan="1" class="tdleft"><code class="func">‣ IsNumberField</code>( <var class="Arg">F</var> )</td><td colspan="1" rowspan="1" class="tdright">( property )</td></tr></tbody></table></div><p xmlns="http://www.w3.org/1999/xhtml">returns <code class="keyw">true</code> if the field <var class="Arg">F</var> is a finite dimensional extension of a prime field in characteristic zero, and <code class="keyw">false</code> otherwise.</p> </body>
+ </html>
\ No newline at end of file
diff --git a/export/mwscrawler/2.html b/export/mwscrawler/2.html
index ca0596f..ce84dcd 100644
--- a/export/mwscrawler/2.html
+++ b/export/mwscrawler/2.html
@@ -1,7 +1,37 @@
<html>
<head>
- <title>TODO</title>
- <meta name="url" content="http://www.gap-system.org/Manuals/doc/ref/TODO#X833C410D85CF96A4"></meta>
+ <title>2.4 Lists and Tables</title>
+ <meta name="url" content="http://www.gap-system.org/Manuals/doc/ref/TODO#X7F10E951789D6EDF"></meta>
</head>
- <body> <h4 xmlns="http://www.w3.org/1999/xhtml">2.3 <span class="Heading">Crossreferencing</span></h4><p xmlns="http://www.w3.org/1999/xhtml">[→ <a shape="rect" href="chapB_mj.html#X7D19CF4782309661"><span class="RefLink">B.8</span></a>]</p><p xmlns="http://www.w3.org/1999/xhtml">In this section we demonstrate various references to parts of this document. Here is a reference to this section: <a shape="rect" href="chap2_mj.html#X833C410D85CF96A4"><span class="RefLink">2.3</span></a>. Here is a reference to chapter <a shape="rect" href="chap1_mj.html#X80E2AD7481DD69D9"><span class="RefLink">1</span></a>, to appendix <a shape="rect" href="chapA_mj.html#X7B53252784137533"><span class="RefLink">A</span></a>, and to subsection <a shape="rect" href="chap1_mj.html#X7E193BD379F58A4C"><span class="RefLink">1.1-1</span></a>.</p><p xmlns="http://www.w3.org/1999/xhtml">We distinguish among others references to functions (see <code class="func">f</code> (<a shape="rect" href="chap1_mj.html#X7FA1D0937FA1D093"><span class="RefLink">1.2-1</span></a>)), to methods with tricky name (see <code class="func">\^\{\}\[\]\<\&</code> (<a shape="rect" href="chap1_mj.html#X822B5C487B29E799"><span class="RefLink">1.2-2</span></a>)), to operations (see <code class="func">MyOperation</code> (<a shape="rect" href="chap1_mj.html#X7D33C2597988F481"><span class="RefLink">1.2-3</span></a>)), to methods (see <code class="func">MyOperation</code> (<a shape="rect" href="chap1_mj.html#X783DCD4E826289D4"><span class="RefLink">1.2-4</span></a>) or <code class="func">MyOperation</code> (<a shape="rect" href="chap1_mj.html#X7A5F4A287D06988C"><span class="RefLink">1.2-5</span></a>)), to filters (see <code class="func">IsBla</code> (<a shape="rect" href="chap1_mj.html#X82954B687D2DF3C2"><span class="RefLink">1.2-6</span></a>)), to properties (see <code class="func">IsBlubb</code> (<a shape="rect" href="chap1_mj.html#X80C364DD7C919CCE"><span class="RefLink">1.2-7</span></a>)), to attributes (see <code class="func">NumberBlobbs</code> (<a shape="rect" href="chap1_mj.html#X8052A45E7F9F054C"><span class="RefLink">1.2-8</span></a>)), to variables (<code class="func">AllBlibbs</code> (<a shape="rect" href="chap1_mj.html#X7C00E05A7DDEF003"><span class="RefLink">1.2-9</span></a>)), to families (see <code class="func">BlibbsFamily</code> (<a shape="rect" href="chap1_mj.html#X7CBC935A8142E374"><span class="RefLink">1.2-10</span></a>)), and to info classes (see <code class="func">InfoBlibbs</code> (<a shape="rect" href="chap1_mj.html#X84D7D77378AD030A"><span class="RefLink">1.2-11</span></a>)).</p><p xmlns="http://www.w3.org/1999/xhtml">There are also references to labels: see <a shape="rect" href="chap2_mj.html#X833C410D85CF96A4"><span class="RefLink">here</span></a>, to other books: see <a shape="rect" href="../../pkg/GAPDoc-1.5.1/doc/chap3_mj.html#X7B76F6F786521F6B"><span class="RefLink">GAPDoc: What is a DTD?</span></a> or <code class="func">IsSubgroup</code> (<a shape="rect" href="../../../doc/ref/chap40_mj.html#X7839D8927E778334"><span class="RefLink">Reference: IsSubgroup</span></a>) in the <strong class="pkg">GAP</strong> reference manual.</p><p xmlns="http://www.w3.org/1999/xhtml">References to sections come in two styles: <a shape="rect" href="chap1_mj.html#X80E2AD7481DD69D9"><span class="RefLink">1</span></a> or <a shape="rect" href="chap1_mj.html#X80E2AD7481DD69D9"><span class="RefLink"><span class="Heading">Sectioning Elements</span></span></a>.</p><p xmlns="http://www.w3.org/1999/xhtml">Another type of cross referencing is bibliography. Here is a citation: <a shape="rect" href="chapBib_mj.html#biBCR1">[CR81, (5.22)]</a> is an interesting lemma.</p><p xmlns="http://www.w3.org/1999/xhtml">There are also URLs:</p><p xmlns="http://www.w3.org/1999/xhtml"><span class="URL"><a shape="rect" href="http://www.math.rwth-aachen.de/">http://www.math.rwth-aachen.de/</a></span></p><p xmlns="http://www.w3.org/1999/xhtml">Email addresses have a special element: <span class="URL"><a shape="rect" href="mailto:Frank.Luebeck@Math.RWTH-Aachen.De">Frank.Luebeck@Math.RWTH-Aachen.De</a></span></p><p xmlns="http://www.w3.org/1999/xhtml">and Homepages another one: <span class="URL"><a shape="rect" href="http://www-groups.mcs.st-and.ac.uk/~neunhoef/">http://www-groups.mcs.st-and.ac.uk/~neunhoef/</a></span></p><p xmlns="http://www.w3.org/1999/xhtml">And here is a link to the <span class="URL"><a shape="rect" href="http://www.math.rwth-aachen.de/~Frank.Luebeck/gap/EDIM/index.html#ARCHS"><strong class="pkg">EDIM</strong> archives</a></span>.</p><p xmlns="http://www.w3.org/1999/xhtml">One can generate index entries as follows (look up the words "TeX-UserGroup", "RWTH", "Aachen, Hauptbahnhof", and "<strong class="pkg">GAP</strong>, <strong class="pkg">GAPDoc</strong>").</p> </body>
+ <body> <h4 xmlns="http://www.w3.org/1999/xhtml">2.4 <span class="Heading">Lists and Tables</span></h4><p xmlns="http://www.w3.org/1999/xhtml">[→ <a shape="rect" href="chapB_mj.html#X7BB822947F626E1A"><span class="RefLink">B.9</span></a>]</p><p xmlns="http://www.w3.org/1999/xhtml">There are</p><ul xmlns="http://www.w3.org/1999/xhtml"><li><p>lists</p>
+
+</li><li><p>enumerations, and</p>
+
+</li><li><p>tables</p>
+
+</li></ul><p xmlns="http://www.w3.org/1999/xhtml">or:</p><ol xmlns="http://www.w3.org/1999/xhtml"><li><p>lists</p>
+
+</li><li><p>enumerations, and</p>
+
+</li><li><p>tables</p>
+
+</li></ol><p xmlns="http://www.w3.org/1999/xhtml">or with marks:</p><dl xmlns="http://www.w3.org/1999/xhtml"><dt><strong class="Mark">lists:</strong></dt><dd><p>not numbered</p>
+
+</dd><dt><strong class="Mark">enumerations:</strong></dt><dd><p>numbered</p>
+
+</dd><dt><strong class="Mark">tables:</strong></dt><dd><p>two-dimensional</p>
+
+</dd></dl><p xmlns="http://www.w3.org/1999/xhtml">Lists can also be nested:</p><ol xmlns="http://www.w3.org/1999/xhtml"><li><ol><li><p>first item of inner enumeration</p>
+
+</li><li><p>second item of inner enumeration</p>
+
+</li></ol>
+</li><li>
+<ul><li><p>first item of inner list</p>
+
+</li><li><p>second item of inner list</p>
+
+</li></ul>
+</li></ol><p xmlns="http://www.w3.org/1999/xhtml">Here is a <em>table</em>:</p><div class="pcenter" xmlns="http://www.w3.org/1999/xhtml"><table class="GAPDocTable"><caption class="GAPDocTable"><b>Table: </b>Prices</caption><tbody><tr><td colspan="1" rowspan="1" class="tdright">Object</td><td colspan="1" rowspan="1" class="tdcenter">Price</td><td colspan="1" rowspan="1" class="tdleft">available</td></tr><tr><td colspan="1" rowspan="1" class="tdright">Shoe</td><td colspan="1" rowspan="1" class="tdcenter">$1,00</td><td colspan="1" rowspan="1" class="tdleft">there</td></tr><tr><td colspan="1" rowspan="1" class="tdright">Hat</td><td colspan="1" rowspan="1" class="tdcenter">$2,00</td><td colspan="1" rowspan="1" class="tdleft">not there</td></tr></tbody></table><br/><p>Â </p><br/>
+</div> </body>
</html>
\ No newline at end of file
diff --git a/export/mwscrawler/20.html b/export/mwscrawler/20.html
new file mode 100644
index 0000000..502ddc6
--- /dev/null
+++ b/export/mwscrawler/20.html
@@ -0,0 +1,14 @@
+<html>
+ <head>
+ <title>60.2-1 Factors</title>
+ <meta name="url" content="http://www.gap-system.org/Manuals/doc/ref/TODO#X7B0AB0FB7A4136C4"></meta>
+ </head>
+ <body> <h5 xmlns="http://www.w3.org/1999/xhtml">60.2-1 Factors</h5><div class="func" xmlns="http://www.w3.org/1999/xhtml"><table width="100%" class="func"><tbody><tr><td colspan="1" rowspan="1" class="tdleft"><code class="func">‣ Factors</code>( <var class="Arg">F</var> )</td><td colspan="1" rowspan="1" class="tdright">( method )</td></tr></tbody></table></div><p xmlns="http://www.w3.org/1999/xhtml">Factoring of polynomials over abelian number fields consisting of cyclotomics works in principle but is not very efficient if the degree of the field extension is large.</p><div class="example" xmlns="http://www.w3.org/1999/xhtml"><pre xml:space="preserve">
+<span class="GAPprompt">gap></span> <span class="GAPinput">x:= Indeterminate( CF(5) );</span>
+x_1
+<span class="GAPprompt">gap></span> <span class="GAPinput">Factors( PolynomialRing( Rationals ), x^5-1 );</span>
+[ x_1-1, x_1^4+x_1^3+x_1^2+x_1+1 ]
+<span class="GAPprompt">gap></span> <span class="GAPinput">Factors( PolynomialRing( CF(5) ), x^5-1 );</span>
+[ x_1-1, x_1+(-E(5)), x_1+(-E(5)^2), x_1+(-E(5)^3), x_1+(-E(5)^4) ]
+</pre></div> </body>
+ </html>
\ No newline at end of file
diff --git a/export/mwscrawler/21.html b/export/mwscrawler/21.html
new file mode 100644
index 0000000..3a4d159
--- /dev/null
+++ b/export/mwscrawler/21.html
@@ -0,0 +1,7 @@
+<html>
+ <head>
+ <title>60.2 Operations for Abelian Number Fields</title>
+ <meta name="url" content="http://www.gap-system.org/Manuals/doc/ref/TODO#X81B5FE06781DB824"></meta>
+ </head>
+ <body> <h4 xmlns="http://www.w3.org/1999/xhtml">60.2 <span class="Heading">Operations for Abelian Number Fields</span></h4><p xmlns="http://www.w3.org/1999/xhtml">For operations for elements of abelian number fields, e.g., <code class="func">Conductor</code> (<a shape="rect" href="chap18_mj.html#X815D6EC57CBA9827"><span class="RefLink">18.1-7</span></a>) or <code class="func">ComplexConjugate</code> (<a shape="rect" href="chap18_mj.html#X7BE001A0811CD599"><span class="RefLink">18.5-2</span></a>), see Chapter <a shape="rect" href="chap18_mj.html#X7DFC03C187DE4841"><span class="RefLink">18</span></a>.</p> </body>
+ </html>
\ No newline at end of file
diff --git a/export/mwscrawler/22.html b/export/mwscrawler/22.html
new file mode 100644
index 0000000..4860475
--- /dev/null
+++ b/export/mwscrawler/22.html
@@ -0,0 +1,58 @@
+<html>
+ <head>
+ <title>60.1-3 GaussianRationals</title>
+ <meta name="url" content="http://www.gap-system.org/Manuals/doc/ref/TODO#X82F53C65802FF551"></meta>
+ </head>
+ <body> <h5 xmlns="http://www.w3.org/1999/xhtml">60.1-3 GaussianRationals</h5><div class="func" xmlns="http://www.w3.org/1999/xhtml"><table width="100%" class="func"><tbody><tr><td colspan="1" rowspan="1" class="tdleft"><code class="func">‣ GaussianRationals</code></td><td colspan="1" rowspan="1" class="tdright">( global variable )</td></tr></tbody></table></div><div class="func" xmlns="http://www.w3.org/1999/xhtml"><table width="100%" class="func"><tbody><tr><td colspan="1" rowspan="1" class="tdleft"><code class="func">‣ IsGaussianRationals</code>( <var class="Arg">obj</var> )</td><td colspan="1" rowspan="1" class="tdright">( category )</td></tr></tbody></table></div><p xmlns="http://www.w3.org/1999/xhtml"><code class="func">GaussianRationals</code> is the field <math id="9015601468690095953" display="inline" alttext="?_{4}=?(\sqrt{{-1}})" class="ltx_Math" xmlns="http://www.w3.org/1998/Math/MathML">
+ <semantics id="p1.1.m1.1a">
+ <mrow xref="p1.1.m1.1.8.cmml" id="p1.1.m1.1.8">
+ <msub xref="p1.1.m1.1.8.1.cmml" id="p1.1.m1.1.8.1">
+ <mi xref="p1.1.m1.1.1.cmml" id="p1.1.m1.1.1" mathvariant="normal">?</mi>
+ <mn xref="p1.1.m1.1.2.1.cmml" id="p1.1.m1.1.2.1">4</mn>
+ </msub>
+ <mo xref="p1.1.m1.1.3.cmml" id="p1.1.m1.1.3">=</mo>
+ <mrow xref="p1.1.m1.1.8.2.cmml" id="p1.1.m1.1.8.2">
+ <mi xref="p1.1.m1.1.4.cmml" id="p1.1.m1.1.4" mathvariant="normal">?</mi>
+ <mo xref="p1.1.m1.1.8.2.1.cmml" id="p1.1.m1.1.8.2.1">â¢</mo>
+ <mrow xref="p1.1.m1.1.6.cmml" id="p1.1.m1.1.8.2.2">
+ <mo id="p1.1.m1.1.5" stretchy="false">(</mo>
+ <msqrt xref="p1.1.m1.1.6.cmml" id="p1.1.m1.1.6">
+ <mrow xref="p1.1.m1.1.6.cmml" id="p1.1.m1.1.6.2">
+ <mo xref="p1.1.m1.1.6.2.1.cmml" id="p1.1.m1.1.6.2.1">-</mo>
+ <mn xref="p1.1.m1.1.6.2.2.cmml" id="p1.1.m1.1.6.2.2">1</mn>
+ </mrow>
+ </msqrt>
+ <mo id="p1.1.m1.1.7" stretchy="false">)</mo>
+ </mrow>
+ </mrow>
+ </mrow>
+ <annotation-xml id="p1.1.m1.1b" encoding="MathML-Content">
+ <apply xref="p1.1.m1.1.8" id="p1.1.m1.1.8.cmml">
+ <eq xref="p1.1.m1.1.3" id="p1.1.m1.1.3.cmml"/>
+ <apply xref="p1.1.m1.1.8.1" id="p1.1.m1.1.8.1.cmml">
+ <csymbol id="p1.1.m1.1.8.1.1.cmml" cd="ambiguous">subscript</csymbol>
+ <ci xref="p1.1.m1.1.1" id="p1.1.m1.1.1.cmml">?</ci>
+ <cn xref="p1.1.m1.1.2.1" id="p1.1.m1.1.2.1.cmml" type="integer">4</cn>
+ </apply>
+ <apply xref="p1.1.m1.1.8.2" id="p1.1.m1.1.8.2.cmml">
+ <times xref="p1.1.m1.1.8.2.1" id="p1.1.m1.1.8.2.1.cmml"/>
+ <ci xref="p1.1.m1.1.4" id="p1.1.m1.1.4.cmml">?</ci>
+ <apply xref="p1.1.m1.1.8.2.2" id="p1.1.m1.1.6.cmml">
+ <root id="p1.1.m1.1.6a.cmml"/>
+ <apply xref="p1.1.m1.1.8.2.2" id="p1.1.m1.1.6.2.cmml">
+ <minus xref="p1.1.m1.1.6.2.1" id="p1.1.m1.1.6.2.1.cmml"/>
+ <cn xref="p1.1.m1.1.6.2.2" id="p1.1.m1.1.6.2.2.cmml" type="integer">1</cn>
+ </apply>
+ </apply>
+ </apply>
+ </apply>
+ </annotation-xml>
+ <annotation id="p1.1.m1.1c" encoding="application/x-tex">?_{4}=?(\sqrt{{-1}})</annotation>
+ </semantics>
+</math> of Gaussian rationals, as a set of cyclotomic numbers, see Chapter <a shape="rect" href="chap18_mj.html#X7DFC03C187DE4841"><span class="RefLink">18</span></a> for basic operations. This field can also be obtained as <code class="code">CF(4)</code> (see <code class="func">CyclotomicField</code> (<a shape="rect" href="chap60_mj.html#X80D21D80850EFA4B"><span class="RefLink">60.1-1</span></a>)).</p><p xmlns="http://www.w3.org/1999/xhtml">The filter <code class="func">IsGaussianRationals</code> returns <code class="keyw">true</code> for the <strong class="pkg">GAP</strong> object <code class="func">GaussianRationals</code>, and <code class="keyw">false</code> for all other <strong class="pkg">GAP</strong> objects.</p><p xmlns="http://www.w3.org/1999/xhtml">(For details about the field of rationals, see Chapter <code class="func">Rationals</code> (<a shape="rect" href="chap17_mj.html#X7B6029D18570C08A"><span class="RefLink">17.1-1</span></a>).)</p><div class="example" xmlns="http://www.w3.org/1999/xhtml"><pre xml:space="preserve">
+<span class="GAPprompt">gap></span> <span class="GAPinput">CF(4) = GaussianRationals;</span>
+true
+<span class="GAPprompt">gap></span> <span class="GAPinput">Sqrt(-1) in GaussianRationals;</span>
+true
+</pre></div> </body>
+ </html>
\ No newline at end of file
diff --git a/export/mwscrawler/23.html b/export/mwscrawler/23.html
new file mode 100644
index 0000000..aefd2ac
--- /dev/null
+++ b/export/mwscrawler/23.html
@@ -0,0 +1,14 @@
+<html>
+ <head>
+ <title>60.1-2 AbelianNumberField</title>
+ <meta name="url" content="http://www.gap-system.org/Manuals/doc/ref/TODO#X80E5AD028143E11E"></meta>
+ </head>
+ <body> <h5 xmlns="http://www.w3.org/1999/xhtml">60.1-2 AbelianNumberField</h5><div class="func" xmlns="http://www.w3.org/1999/xhtml"><table width="100%" class="func"><tbody><tr><td colspan="1" rowspan="1" class="tdleft"><code class="func">‣ AbelianNumberField</code>( <var class="Arg">n</var>, <var class="Arg">stab</var> )</td><td colspan="1" rowspan="1" class="tdright">( function )</td></tr></tbody></table></div><div class="func" xmlns="http://www.w3.org/1999/xhtml"><table width="100%" class="func"><tbody><tr><td colspan="1" rowspan="1" class="tdleft"><code class="func">‣ NF</code>( <var class="Arg">n</var>, <var class="Arg">stab</var> )</td><td colspan="1" rowspan="1" class="tdright">( function )</td></tr></tbody></table></div><p xmlns="http://www.w3.org/1999/xhtml">For a positive integer <var class="Arg">n</var> and a list <var class="Arg">stab</var> of prime residues modulo <var class="Arg">n</var>, <code class="func">AbelianNumberField</code> returns the fixed field of the group described by <var class="Arg">stab</var> (cf. <code class="func">GaloisStabilizer</code> (<a shape="rect" href="chap60_mj.html#X87E7313D8070B9CC"><span class="RefLink">60.2-5</span></a>)), in the <var class="Arg">n</var>-th cyclotomic field. <code class="func">AbelianNumberField</code> is mainly thought for internal use and for printing fields in a standard way; <code class="func">Field</code> (<a shape="rect" href="chap58_mj.html#X871AA7D58263E9AC"><span class="RefLink">58.1-3</span></a>) (cf. also <a shape="rect" href="chap60_mj.html#X81B5FE06781DB824"><span class="RefLink">60.2</span></a>) is probably more suitable if one knows generators of the field in question.</p><p xmlns="http://www.w3.org/1999/xhtml"><code class="func">AbelianNumberField</code> can be abbreviated to <code class="func">NF</code>, this form is used also when <strong class="pkg">GAP</strong> prints abelian number fields.</p><p xmlns="http://www.w3.org/1999/xhtml">Fields constructed with <code class="func">NF</code> are stored in the global list <code class="code">ABELIAN_NUMBER_FIELDS</code>, so repeated calls of <code class="func">NF</code> just fetch these field objects after they have been created once.</p><div class="example" xmlns="http://www.w3.org/1999/xhtml"><pre xml:space="preserve">
+<span class="GAPprompt">gap></span> <span class="GAPinput">NF( 7, [ 1 ] );</span>
+CF(7)
+<span class="GAPprompt">gap></span> <span class="GAPinput">f:= NF( 7, [ 1, 2 ] ); Sqrt(-7); Sqrt(-7) in f;</span>
+NF(7,[ 1, 2, 4 ])
+E(7)+E(7)^2-E(7)^3+E(7)^4-E(7)^5-E(7)^6
+true
+</pre></div> </body>
+ </html>
\ No newline at end of file
diff --git a/export/mwscrawler/24.html b/export/mwscrawler/24.html
new file mode 100644
index 0000000..748ea87
--- /dev/null
+++ b/export/mwscrawler/24.html
@@ -0,0 +1,29 @@
+<html>
+ <head>
+ <title>60.1-1 CyclotomicField</title>
+ <meta name="url" content="http://www.gap-system.org/Manuals/doc/ref/TODO#X80D21D80850EFA4B"></meta>
+ </head>
+ <body> <h5 xmlns="http://www.w3.org/1999/xhtml">60.1-1 CyclotomicField</h5><div class="func" xmlns="http://www.w3.org/1999/xhtml"><table width="100%" class="func"><tbody><tr><td colspan="1" rowspan="1" class="tdleft"><code class="func">‣ CyclotomicField</code>( [<var class="Arg">subfield</var>, ]<var class="Arg">n</var> )</td><td colspan="1" rowspan="1" class="tdright">( function )</td></tr></tbody></table></div><div class="func" xmlns="http://www.w3.org/1999/xhtml"><table width="100%" class="func"><tbody><tr><td colspan="1" rowspan="1" class="tdleft"><code class="func">‣ CyclotomicField</code>( [<var class="Arg">subfield</var>, ]<var class="Arg">gens</var> )</td><td colspan="1" rowspan="1" class="tdright">( function )</td></tr></tbody></table></div><div class="func" xmlns="http://www.w3.org/1999/xhtml"><table width="100%" class="func"><tbody><tr><td colspan="1" rowspan="1" class="tdleft"><code class="func">‣ CF</code>( [<var class="Arg">subfield</var>, ]<var class="Arg">n</var> )</td><td colspan="1" rowspan="1" class="tdright">( function )</td></tr></tbody></table></div><div class="func" xmlns="http://www.w3.org/1999/xhtml"><table width="100%" class="func"><tbody><tr><td colspan="1" rowspan="1" class="tdleft"><code class="func">‣ CF</code>( [<var class="Arg">subfield</var>, ]<var class="Arg">gens</var> )</td><td colspan="1" rowspan="1" class="tdright">( function )</td></tr></tbody></table></div><p xmlns="http://www.w3.org/1999/xhtml">The first version creates the <var class="Arg">n</var>-th cyclotomic field <math id="2016962131100400854" display="inline" alttext="?_{n}" class="ltx_Math" xmlns="http://www.w3.org/1998/Math/MathML">
+ <semantics id="p1.1.m1.1a">
+ <msub xref="p1.1.m1.1.3.cmml" id="p1.1.m1.1.3">
+ <mi xref="p1.1.m1.1.1.cmml" id="p1.1.m1.1.1" mathvariant="normal">?</mi>
+ <mi xref="p1.1.m1.1.2.1.cmml" id="p1.1.m1.1.2.1">n</mi>
+ </msub>
+ <annotation-xml id="p1.1.m1.1b" encoding="MathML-Content">
+ <apply xref="p1.1.m1.1.3" id="p1.1.m1.1.3.cmml">
+ <csymbol id="p1.1.m1.1.3.1.cmml" cd="ambiguous">subscript</csymbol>
+ <ci xref="p1.1.m1.1.1" id="p1.1.m1.1.1.cmml">?</ci>
+ <ci xref="p1.1.m1.1.2.1" id="p1.1.m1.1.2.1.cmml">ð‘›</ci>
+ </apply>
+ </annotation-xml>
+ <annotation id="p1.1.m1.1c" encoding="application/x-tex">?_{n}</annotation>
+ </semantics>
+</math>. The second version creates the smallest cyclotomic field containing the elements in the list <var class="Arg">gens</var>. In both cases the field can be generated as an extension of a designated subfield <var class="Arg">subfield</var> (cf. <a shape="rect" href="chap60_mj.html#X7D2421AC8491D2BE"><span class="RefLink">60.3</span></a>).</p><p xmlns="http://www.w3.org/1999/xhtml"><code class="func">CyclotomicField</code> can be abbreviated to <code class="func">CF</code>, this form is used also when <strong class="pkg">GAP</strong> prints cyclotomic fields.</p><p xmlns="http://www.w3.org/1999/xhtml">Fields constructed with the one argument version of <code class="func">CF</code> are stored in the global list <code class="code">CYCLOTOMIC_FIELDS</code>, so repeated calls of <code class="func">CF</code> just fetch these field objects after they have been created once.</p><div class="example" xmlns="http://www.w3.org/1999/xhtml"><pre xml:space="preserve">
+<span class="GAPprompt">gap></span> <span class="GAPinput">CyclotomicField( 5 ); CyclotomicField( [ Sqrt(3) ] );</span>
+CF(5)
+CF(12)
+<span class="GAPprompt">gap></span> <span class="GAPinput">CF( CF(3), 12 ); CF( CF(4), [ Sqrt(7) ] );</span>
+AsField( CF(3), CF(12) )
+AsField( GaussianRationals, CF(28) )
+</pre></div> </body>
+ </html>
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diff --git a/export/mwscrawler/25.html b/export/mwscrawler/25.html
new file mode 100644
index 0000000..4f477b9
--- /dev/null
+++ b/export/mwscrawler/25.html
@@ -0,0 +1,7 @@
+<html>
+ <head>
+ <title>60.1 Construction of Abelian Number Fields</title>
+ <meta name="url" content="http://www.gap-system.org/Manuals/doc/ref/TODO#X7D4E43E5799753B5"></meta>
+ </head>
+ <body> <h4 xmlns="http://www.w3.org/1999/xhtml">60.1 <span class="Heading">Construction of Abelian Number Fields</span></h4><p xmlns="http://www.w3.org/1999/xhtml">Besides the usual construction using <code class="func">Field</code> (<a shape="rect" href="chap58_mj.html#X871AA7D58263E9AC"><span class="RefLink">58.1-3</span></a>) or <code class="func">DefaultField</code> (<a shape="rect" href="chap18_mj.html#X7FE3D5637B5485D0"><span class="RefLink">18.1-16</span></a>) (see <code class="func">DefaultField</code> (<a shape="rect" href="chap18_mj.html#X7FE3D5637B5485D0"><span class="RefLink">18.1-16</span></a>)), abelian number fields consisting of cyclotomics can be created with <code class="func">CyclotomicField</code> (<a shape="rect" href="chap60_mj.html#X80D21D80850EFA4B"><span class="RefLink">60.1-1</span></a>) and <code class="func">AbelianNumberField</code> (<a shape="rect" href="chap60_mj.html#X80E5AD028143E11E"><span class="RefLink">60.1-2</span></a>).</p> </body>
+ </html>
\ No newline at end of file
diff --git a/export/mwscrawler/26.html b/export/mwscrawler/26.html
new file mode 100644
index 0000000..ac2a8c8
--- /dev/null
+++ b/export/mwscrawler/26.html
@@ -0,0 +1,73 @@
+<html>
+ <head>
+ <title>TODO</title>
+ <meta name="url" content="http://www.gap-system.org/Manuals/doc/ref/TODO#X80510B5880521FDC"></meta>
+ </head>
+ <body> <div class="ChapSects" xmlns="http://www.w3.org/1999/xhtml"><a shape="rect" href="chap60_mj.html#X80510B5880521FDC">60 <span class="Heading">Abelian Number Fields</span></a>
+<div class="ContSect"><span class="tocline"><span class="nocss">Â </span><a shape="rect" href="chap60_mj.html#X7D4E43E5799753B5">60.1 <span class="Heading">Construction of Abelian Number Fields</span></a>
+</span>
+<div class="ContSSBlock">
+<span class="ContSS"><br/><span class="nocss">Â Â </span><a shape="rect" href="chap60_mj.html#X80D21D80850EFA4B">60.1-1 CyclotomicField</a></span>
+<span class="ContSS"><br/><span class="nocss">Â Â </span><a shape="rect" href="chap60_mj.html#X80E5AD028143E11E">60.1-2 AbelianNumberField</a></span>
+<span class="ContSS"><br/><span class="nocss">Â Â </span><a shape="rect" href="chap60_mj.html#X82F53C65802FF551">60.1-3 GaussianRationals</a></span>
+</div></div>
+<div class="ContSect"><span class="tocline"><span class="nocss">Â </span><a shape="rect" href="chap60_mj.html#X81B5FE06781DB824">60.2 <span class="Heading">Operations for Abelian Number Fields</span></a>
+</span>
+<div class="ContSSBlock">
+<span class="ContSS"><br/><span class="nocss">Â Â </span><a shape="rect" href="chap60_mj.html#X7B0AB0FB7A4136C4">60.2-1 Factors</a></span>
+<span class="ContSS"><br/><span class="nocss">Â Â </span><a shape="rect" href="chap60_mj.html#X87D78F5E875F2E8A">60.2-2 IsNumberField</a></span>
+<span class="ContSS"><br/><span class="nocss">Â Â </span><a shape="rect" href="chap60_mj.html#X7D202D707D5708FA">60.2-3 IsAbelianNumberField</a></span>
+<span class="ContSS"><br/><span class="nocss">Â Â </span><a shape="rect" href="chap60_mj.html#X84CAE4627F0CD639">60.2-4 IsCyclotomicField</a></span>
+<span class="ContSS"><br/><span class="nocss">Â Â </span><a shape="rect" href="chap60_mj.html#X87E7313D8070B9CC">60.2-5 GaloisStabilizer</a></span>
+</div></div>
+<div class="ContSect"><span class="tocline"><span class="nocss">Â </span><a shape="rect" href="chap60_mj.html#X7D2421AC8491D2BE">60.3 <span class="Heading">Integral Bases of Abelian Number Fields</span></a>
+</span>
+<div class="ContSSBlock">
+<span class="ContSS"><br/><span class="nocss">Â Â </span><a shape="rect" href="chap60_mj.html#X7F52BEA0862E06F2">60.3-1 ZumbroichBase</a></span>
+<span class="ContSS"><br/><span class="nocss">Â Â </span><a shape="rect" href="chap60_mj.html#X87DB9C2C858B722A">60.3-2 LenstraBase</a></span>
+</div></div>
+<div class="ContSect"><span class="tocline"><span class="nocss">Â </span><a shape="rect" href="chap60_mj.html#X7E4AB4B17C7BA10C">60.4 <span class="Heading">Galois Groups of Abelian Number Fields</span></a>
+</span>
+<div class="ContSSBlock">
+<span class="ContSS"><br/><span class="nocss">Â Â </span><a shape="rect" href="chap60_mj.html#X7B55A90582E818F3">60.4-1 GaloisGroup</a></span>
+<span class="ContSS"><br/><span class="nocss">Â Â </span><a shape="rect" href="chap60_mj.html#X8643D4B47A827D9D">60.4-2 ANFAutomorphism</a></span>
+</div></div>
+<div class="ContSect"><span class="tocline"><span class="nocss">Â </span><a shape="rect" href="chap60_mj.html#X85E9E90D7FE877CC">60.5 <span class="Heading">Gaussians</span></a>
+</span>
+<div class="ContSSBlock">
+<span class="ContSS"><br/><span class="nocss">Â Â </span><a shape="rect" href="chap60_mj.html#X80BD5EAB879F096E">60.5-1 GaussianIntegers</a></span>
+<span class="ContSS"><br/><span class="nocss">Â Â </span><a shape="rect" href="chap60_mj.html#X7BFD33D27BFB7C5A">60.5-2 IsGaussianIntegers</a></span>
+</div></div>
+</div><h3 xmlns="http://www.w3.org/1999/xhtml">60 <span class="Heading">Abelian Number Fields</span></h3><p xmlns="http://www.w3.org/1999/xhtml">An <em>abelian number field</em> is a field in characteristic zero that is a finite dimensional normal extension of its prime field such that the Galois group is abelian. In <strong class="pkg">GAP</strong>, one implementation of abelian number fields is given by fields of cyclotomic numbers (see Chapter <a shape="rect" href="chap18_mj.html#X7DFC03C187DE4841"><span class="RefLink">18</span></a>). Note that abelian number fields can also be constructed with the more general <code class="func">AlgebraicExtension</code> (<a shape="rect" href="chap67_mj.html#X7CDA90537D2BAC8A"><span class="RefLink">67.1-1</span></a>), a discussion of advantages and disadvantages can be found in <a shape="rect" href="chap18_mj.html#X8557FC2D7ACD6105"><span class="RefLink">18.6</span></a>. The functions described in this chapter have been developed for fields whose elements are in the filter <code class="func">IsCyclotomic</code> (<a shape="rect" href="chap18_mj.html#X841C425281A6F775"><span class="RefLink">18.1-3</span></a>), they may or may not work well for abelian number fields consisting of other kinds of elements.</p><p xmlns="http://www.w3.org/1999/xhtml">Throughout this chapter, <math id="8586465557677769213" display="inline" alttext="?_{n}" class="ltx_Math" xmlns="http://www.w3.org/1998/Math/MathML">
+ <semantics id="p1.1.m1.1a">
+ <msub xref="p1.1.m1.1.3.cmml" id="p1.1.m1.1.3">
+ <mi xref="p1.1.m1.1.1.cmml" id="p1.1.m1.1.1" mathvariant="normal">?</mi>
+ <mi xref="p1.1.m1.1.2.1.cmml" id="p1.1.m1.1.2.1">n</mi>
+ </msub>
+ <annotation-xml id="p1.1.m1.1b" encoding="MathML-Content">
+ <apply xref="p1.1.m1.1.3" id="p1.1.m1.1.3.cmml">
+ <csymbol id="p1.1.m1.1.3.1.cmml" cd="ambiguous">subscript</csymbol>
+ <ci xref="p1.1.m1.1.1" id="p1.1.m1.1.1.cmml">?</ci>
+ <ci xref="p1.1.m1.1.2.1" id="p1.1.m1.1.2.1.cmml">ð‘›</ci>
+ </apply>
+ </annotation-xml>
+ <annotation id="p1.1.m1.1c" encoding="application/x-tex">?_{n}</annotation>
+ </semantics>
+</math> will denote the cyclotomic field generated by the field <math id="8103627879763627883" display="inline" alttext="?" class="ltx_Math" xmlns="http://www.w3.org/1998/Math/MathML">
+ <semantics id="p1.1.m1.1a">
+ <mi xref="p1.1.m1.1.1.cmml" id="p1.1.m1.1.1" mathvariant="normal">?</mi>
+ <annotation-xml id="p1.1.m1.1b" encoding="MathML-Content">
+ <ci xref="p1.1.m1.1.1" id="p1.1.m1.1.1.cmml">?</ci>
+ </annotation-xml>
+ <annotation id="p1.1.m1.1c" encoding="application/x-tex">?</annotation>
+ </semantics>
+</math> of rationals together with <math id="-4895042398210646987" display="inline" alttext="n" class="ltx_Math" xmlns="http://www.w3.org/1998/Math/MathML">
+ <semantics id="p1.1.m1.1a">
+ <mi xref="p1.1.m1.1.1.cmml" id="p1.1.m1.1.1">n</mi>
+ <annotation-xml id="p1.1.m1.1b" encoding="MathML-Content">
+ <ci xref="p1.1.m1.1.1" id="p1.1.m1.1.1.cmml">ð‘›</ci>
+ </annotation-xml>
+ <annotation id="p1.1.m1.1c" encoding="application/x-tex">n</annotation>
+ </semantics>
+</math>-th roots of unity.</p><p xmlns="http://www.w3.org/1999/xhtml">In <a shape="rect" href="chap60_mj.html#X7D4E43E5799753B5"><span class="RefLink">60.1</span></a>, constructors for abelian number fields are described, <a shape="rect" href="chap60_mj.html#X81B5FE06781DB824"><span class="RefLink">60.2</span></a> introduces operations for abelian number fields, <a shape="rect" href="chap60_mj.html#X7D2421AC8491D2BE"><span class="RefLink">60.3</span></a> deals with the vector space structure of abelian number fields, and <a shape="rect" href="chap60_mj.html#X7E4AB4B17C7BA10C"><span class="RefLink">60.4</span></a> describes field automorphisms of abelian number fields,</p> </body>
+ </html>
\ No newline at end of file
diff --git a/export/mwscrawler/3.html b/export/mwscrawler/3.html
index 7db77f5..4c1dec8 100644
--- a/export/mwscrawler/3.html
+++ b/export/mwscrawler/3.html
@@ -1,301 +1,7 @@
<html>
<head>
- <title>TODO</title>
- <meta name="url" content="http://www.gap-system.org/Manuals/doc/ref/TODO#X7AA5BF0279938BE0"></meta>
+ <title>2.3 Crossreferencing</title>
+ <meta name="url" content="http://www.gap-system.org/Manuals/doc/ref/TODO#X833C410D85CF96A4"></meta>
</head>
- <body> <h4 xmlns="http://www.w3.org/1999/xhtml">2.2 <span class="Heading">Formulae</span></h4><p xmlns="http://www.w3.org/1999/xhtml">[→ <a shape="rect" href="chapB_mj.html#X8516FAA07A95BBB5"><span class="RefLink">B.7</span></a>]</p><p xmlns="http://www.w3.org/1999/xhtml">There are three types of formulae.</p><p xmlns="http://www.w3.org/1999/xhtml">The first is the <em>normal math mode</em> of LaTeX: <math id="-769258759535310604" display="inline" alttext="b_{i}\cdot b_{j}=\sum_{k=1}^{d}h_{ijk}b_{k}" class="ltx_Math" xmlns="http://www.w3.org/1998/Math/MathML">
- <semantics id="p1.1.m1.1a">
- <mrow xref="p1.1.m1.1.14.cmml" id="p1.1.m1.1.14">
- <mrow xref="p1.1.m1.1.14.1.cmml" id="p1.1.m1.1.14.1">
- <msub xref="p1.1.m1.1.14.1.1.cmml" id="p1.1.m1.1.14.1.1">
- <mi xref="p1.1.m1.1.1.cmml" id="p1.1.m1.1.1">b</mi>
- <mi xref="p1.1.m1.1.2.1.cmml" id="p1.1.m1.1.2.1">i</mi>
- </msub>
- <mo xref="p1.1.m1.1.3.cmml" id="p1.1.m1.1.3">â‹…</mo>
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- <mi xref="p1.1.m1.1.5.1.cmml" id="p1.1.m1.1.5.1">j</mi>
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- </mrow>
- <mo xref="p1.1.m1.1.6.cmml" id="p1.1.m1.1.6">=</mo>
- <mrow xref="p1.1.m1.1.14.2.cmml" id="p1.1.m1.1.14.2">
- <msubsup xref="p1.1.m1.1.14.2.1.cmml" id="p1.1.m1.1.14.2.1">
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- <mi xref="p1.1.m1.1.11.1.3.cmml" id="p1.1.m1.1.11.1.3">k</mi>
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- <mo xref="p1.1.m1.1.14.2.2.1.cmml" id="p1.1.m1.1.14.2.2.1">â¢</mo>
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- <mi xref="p1.1.m1.1.12.cmml" id="p1.1.m1.1.12">b</mi>
- <mi xref="p1.1.m1.1.13.1.cmml" id="p1.1.m1.1.13.1">k</mi>
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- <ci xref="p1.1.m1.1.3" id="p1.1.m1.1.3.cmml">â‹…</ci>
- <apply xref="p1.1.m1.1.14.1.1" id="p1.1.m1.1.14.1.1.cmml">
- <csymbol id="p1.1.m1.1.14.1.1.1.cmml" cd="ambiguous">subscript</csymbol>
- <ci xref="p1.1.m1.1.1" id="p1.1.m1.1.1.cmml">ð‘</ci>
- <ci xref="p1.1.m1.1.2.1" id="p1.1.m1.1.2.1.cmml">ð‘–</ci>
- </apply>
- <apply xref="p1.1.m1.1.14.1.2" id="p1.1.m1.1.14.1.2.cmml">
- <csymbol id="p1.1.m1.1.14.1.2.1.cmml" cd="ambiguous">subscript</csymbol>
- <ci xref="p1.1.m1.1.4" id="p1.1.m1.1.4.cmml">ð‘</ci>
- <ci xref="p1.1.m1.1.5.1" id="p1.1.m1.1.5.1.cmml">ð‘—</ci>
- </apply>
- </apply>
- <apply xref="p1.1.m1.1.14.2" id="p1.1.m1.1.14.2.cmml">
- <apply xref="p1.1.m1.1.14.2.1" id="p1.1.m1.1.14.2.1.cmml">
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- <sum xref="p1.1.m1.1.7" id="p1.1.m1.1.7.cmml"/>
- <apply xref="p1.1.m1.1.8.1" id="p1.1.m1.1.8.1.cmml">
- <eq xref="p1.1.m1.1.8.1.2" id="p1.1.m1.1.8.1.2.cmml"/>
- <ci xref="p1.1.m1.1.8.1.1" id="p1.1.m1.1.8.1.1.cmml">ð‘˜</ci>
- <cn xref="p1.1.m1.1.8.1.3" id="p1.1.m1.1.8.1.3.cmml" type="integer">1</cn>
- </apply>
- </apply>
- <ci xref="p1.1.m1.1.9.1" id="p1.1.m1.1.9.1.cmml">ð‘‘</ci>
- </apply>
- <apply xref="p1.1.m1.1.14.2.2" id="p1.1.m1.1.14.2.2.cmml">
- <times xref="p1.1.m1.1.14.2.2.1" id="p1.1.m1.1.14.2.2.1.cmml"/>
- <apply xref="p1.1.m1.1.14.2.2.2" id="p1.1.m1.1.14.2.2.2.cmml">
- <csymbol id="p1.1.m1.1.14.2.2.2.1.cmml" cd="ambiguous">subscript</csymbol>
- <ci xref="p1.1.m1.1.10" id="p1.1.m1.1.10.cmml">â„Ž</ci>
- <apply xref="p1.1.m1.1.11.1" id="p1.1.m1.1.11.1.cmml">
- <times xref="p1.1.m1.1.11.1.4" id="p1.1.m1.1.11.1.4.cmml"/>
- <ci xref="p1.1.m1.1.11.1.1" id="p1.1.m1.1.11.1.1.cmml">ð‘–</ci>
- <ci xref="p1.1.m1.1.11.1.2" id="p1.1.m1.1.11.1.2.cmml">ð‘—</ci>
- <ci xref="p1.1.m1.1.11.1.3" id="p1.1.m1.1.11.1.3.cmml">ð‘˜</ci>
- </apply>
- </apply>
- <apply xref="p1.1.m1.1.14.2.2.3" id="p1.1.m1.1.14.2.2.3.cmml">
- <csymbol id="p1.1.m1.1.14.2.2.3.1.cmml" cd="ambiguous">subscript</csymbol>
- <ci xref="p1.1.m1.1.12" id="p1.1.m1.1.12.cmml">ð‘</ci>
- <ci xref="p1.1.m1.1.13.1" id="p1.1.m1.1.13.1.cmml">ð‘˜</ci>
- </apply>
- </apply>
- </apply>
- </apply>
- </annotation-xml>
- <annotation id="p1.1.m1.1c" encoding="application/x-tex">b_{i}\cdot b_{j}=\sum_{k=1}^{d}h_{ijk}b_{k}</annotation>
- </semantics>
-</math>. Then there are <em>displayed formulae</em>:</p><p class="center" xmlns="http://www.w3.org/1999/xhtml">\[
- \Longrightarrow \quad \left(\sum_{i=1}^d x_i b_i \right) \cdot
- \left(\sum_{j=1}^d y_j b_j \right) =
- \sum_{k=1}^d \left( \sum_{i,j} x_i y_j h_{ijk} \right) b_k
-\]</p><p xmlns="http://www.w3.org/1999/xhtml">If possible, use the <code class="code">Alt</code> element to specify a better readable text version of such a formula as in the following example:</p><pre xml:space="preserve" class="normal" xmlns="http://www.w3.org/1999/xhtml">
-
- d d d
- ----- ----- ----- -----
- \ \ \ \
- ==> ( ) x_i b_i )( ) y_j b_j ) = ) ( ) x_i y_j h_ijk ) b_k
- / / / /
- ----- ----- ----- -----
- i = 1 j = 1 k = 1 i,j
-
-</pre><p xmlns="http://www.w3.org/1999/xhtml">For small formulae without "difficult" parts use the <code class="code">M</code> element: <math id="-9181387147039780882" display="inline" alttext="b_{i}" class="ltx_Math" xmlns="http://www.w3.org/1998/Math/MathML">
- <semantics id="p1.1.m1.1a">
- <msub xref="p1.1.m1.1.3.cmml" id="p1.1.m1.1.3">
- <mi xref="p1.1.m1.1.1.cmml" id="p1.1.m1.1.1">b</mi>
- <mi xref="p1.1.m1.1.2.1.cmml" id="p1.1.m1.1.2.1">i</mi>
- </msub>
- <annotation-xml id="p1.1.m1.1b" encoding="MathML-Content">
- <apply xref="p1.1.m1.1.3" id="p1.1.m1.1.3.cmml">
- <csymbol id="p1.1.m1.1.3.1.cmml" cd="ambiguous">subscript</csymbol>
- <ci xref="p1.1.m1.1.1" id="p1.1.m1.1.1.cmml">ð‘</ci>
- <ci xref="p1.1.m1.1.2.1" id="p1.1.m1.1.2.1.cmml">ð‘–</ci>
- </apply>
- </annotation-xml>
- <annotation id="p1.1.m1.1c" encoding="application/x-tex">b_{i}</annotation>
- </semantics>
-</math>, <math id="4919001157076060698" display="inline" alttext="x^{2}" class="ltx_Math" xmlns="http://www.w3.org/1998/Math/MathML">
- <semantics id="p1.1.m1.1a">
- <msup xref="p1.1.m1.1.3.cmml" id="p1.1.m1.1.3">
- <mi xref="p1.1.m1.1.1.cmml" id="p1.1.m1.1.1">x</mi>
- <mn xref="p1.1.m1.1.2.1.cmml" id="p1.1.m1.1.2.1">2</mn>
- </msup>
- <annotation-xml id="p1.1.m1.1b" encoding="MathML-Content">
- <apply xref="p1.1.m1.1.3" id="p1.1.m1.1.3.cmml">
- <csymbol id="p1.1.m1.1.3.1.cmml" cd="ambiguous">superscript</csymbol>
- <ci xref="p1.1.m1.1.1" id="p1.1.m1.1.1.cmml">ð‘¥</ci>
- <cn xref="p1.1.m1.1.2.1" id="p1.1.m1.1.2.1.cmml" type="integer">2</cn>
- </apply>
- </annotation-xml>
- <annotation id="p1.1.m1.1c" encoding="application/x-tex">x^{2}</annotation>
- </semantics>
-</math>, <math id="3842779778635680353" display="inline" alttext="x^{2}+2x+1=(x+1)^{2}" class="ltx_Math" xmlns="http://www.w3.org/1998/Math/MathML">
- <semantics id="p1.1.m1.1a">
- <mrow xref="p1.1.m1.1.15.cmml" id="p1.1.m1.1.15">
- <mrow xref="p1.1.m1.1.15.1.cmml" id="p1.1.m1.1.15.1">
- <msup xref="p1.1.m1.1.15.1.1.cmml" id="p1.1.m1.1.15.1.1">
- <mi xref="p1.1.m1.1.1.cmml" id="p1.1.m1.1.1">x</mi>
- <mn xref="p1.1.m1.1.2.1.cmml" id="p1.1.m1.1.2.1">2</mn>
- </msup>
- <mo xref="p1.1.m1.1.3.cmml" id="p1.1.m1.1.3">+</mo>
- <mrow xref="p1.1.m1.1.15.1.2.cmml" id="p1.1.m1.1.15.1.2">
- <mn xref="p1.1.m1.1.4.cmml" id="p1.1.m1.1.4">2</mn>
- <mo xref="p1.1.m1.1.15.1.2.1.cmml" id="p1.1.m1.1.15.1.2.1">â¢</mo>
- <mi xref="p1.1.m1.1.5.cmml" id="p1.1.m1.1.5">x</mi>
- </mrow>
- <mo xref="p1.1.m1.1.3.cmml" id="p1.1.m1.1.3a">+</mo>
- <mn xref="p1.1.m1.1.7.cmml" id="p1.1.m1.1.7">1</mn>
- </mrow>
- <mo xref="p1.1.m1.1.8.cmml" id="p1.1.m1.1.8">=</mo>
- <msup xref="p1.1.m1.1.15.2.cmml" id="p1.1.m1.1.15.2">
- <mrow xref="p1.1.m1.1.15.2.2.2.cmml" id="p1.1.m1.1.15.2.2">
- <mo id="p1.1.m1.1.9" stretchy="false">(</mo>
- <mrow xref="p1.1.m1.1.15.2.2.2.cmml" id="p1.1.m1.1.15.2.2.2">
- <mi xref="p1.1.m1.1.10.cmml" id="p1.1.m1.1.10">x</mi>
- <mo xref="p1.1.m1.1.11.cmml" id="p1.1.m1.1.11">+</mo>
- <mn xref="p1.1.m1.1.12.cmml" id="p1.1.m1.1.12">1</mn>
- </mrow>
- <mo id="p1.1.m1.1.13" stretchy="false">)</mo>
- </mrow>
- <mn xref="p1.1.m1.1.14.1.cmml" id="p1.1.m1.1.14.1">2</mn>
- </msup>
- </mrow>
- <annotation-xml id="p1.1.m1.1b" encoding="MathML-Content">
- <apply xref="p1.1.m1.1.15" id="p1.1.m1.1.15.cmml">
- <eq xref="p1.1.m1.1.8" id="p1.1.m1.1.8.cmml"/>
- <apply xref="p1.1.m1.1.15.1" id="p1.1.m1.1.15.1.cmml">
- <plus xref="p1.1.m1.1.3" id="p1.1.m1.1.3.cmml"/>
- <apply xref="p1.1.m1.1.15.1.1" id="p1.1.m1.1.15.1.1.cmml">
- <csymbol id="p1.1.m1.1.15.1.1.1.cmml" cd="ambiguous">superscript</csymbol>
- <ci xref="p1.1.m1.1.1" id="p1.1.m1.1.1.cmml">ð‘¥</ci>
- <cn xref="p1.1.m1.1.2.1" id="p1.1.m1.1.2.1.cmml" type="integer">2</cn>
- </apply>
- <apply xref="p1.1.m1.1.15.1.2" id="p1.1.m1.1.15.1.2.cmml">
- <times xref="p1.1.m1.1.15.1.2.1" id="p1.1.m1.1.15.1.2.1.cmml"/>
- <cn xref="p1.1.m1.1.4" id="p1.1.m1.1.4.cmml" type="integer">2</cn>
- <ci xref="p1.1.m1.1.5" id="p1.1.m1.1.5.cmml">ð‘¥</ci>
- </apply>
- <cn xref="p1.1.m1.1.7" id="p1.1.m1.1.7.cmml" type="integer">1</cn>
- </apply>
- <apply xref="p1.1.m1.1.15.2" id="p1.1.m1.1.15.2.cmml">
- <csymbol id="p1.1.m1.1.15.2.1.cmml" cd="ambiguous">superscript</csymbol>
- <apply xref="p1.1.m1.1.15.2.2" id="p1.1.m1.1.15.2.2.2.cmml">
- <plus xref="p1.1.m1.1.11" id="p1.1.m1.1.11.cmml"/>
- <ci xref="p1.1.m1.1.10" id="p1.1.m1.1.10.cmml">ð‘¥</ci>
- <cn xref="p1.1.m1.1.12" id="p1.1.m1.1.12.cmml" type="integer">1</cn>
- </apply>
- <cn xref="p1.1.m1.1.14.1" id="p1.1.m1.1.14.1.cmml" type="integer">2</cn>
- </apply>
- </apply>
- </annotation-xml>
- <annotation id="p1.1.m1.1c" encoding="application/x-tex">x^{2}+2x+1=(x+1)^{2}</annotation>
- </semantics>
-</math>. Note that here whitespace matters for text (or HTML) output.</p><p xmlns="http://www.w3.org/1999/xhtml">Here are two formulae containing less than characters which are special characters for XML: <math id="1913985833526992080" display="inline" alttext="a&lt;b&lt;c&lt;d" class="ltx_Math" xmlns="http://www.w3.org/1998/Math/MathML">
- <semantics id="p1.1.m1.1a">
- <mrow xref="p1.1.m1.1.17.1.cmml" id="p1.1.m1.1.17">
- <mrow xref="p1.1.m1.1.17.1.cmml" id="p1.1.m1.1.17.2">
- <mi xref="p1.1.m1.1.1.cmml" id="p1.1.m1.1.1">a</mi>
- <mo xref="p1.1.m1.1.17.2.1.cmml" id="p1.1.m1.1.17.2.1">â¢</mo>
- <mi xref="p1.1.m1.1.2.cmml" id="p1.1.m1.1.2" mathvariant="normal">&</mi>
- <mo xref="p1.1.m1.1.17.2.1.cmml" id="p1.1.m1.1.17.2.1a">â¢</mo>
- <mi xref="p1.1.m1.1.3.cmml" id="p1.1.m1.1.3">l</mi>
- <mo xref="p1.1.m1.1.17.2.1.cmml" id="p1.1.m1.1.17.2.1b">â¢</mo>
- <mi xref="p1.1.m1.1.4.cmml" id="p1.1.m1.1.4">t</mi>
- </mrow>
- <mo id="p1.1.m1.1.5">;</mo>
- <mrow xref="p1.1.m1.1.17.1.cmml" id="p1.1.m1.1.17.3">
- <mi xref="p1.1.m1.1.6.cmml" id="p1.1.m1.1.6">b</mi>
- <mo xref="p1.1.m1.1.17.3.1.cmml" id="p1.1.m1.1.17.3.1">â¢</mo>
- <mi xref="p1.1.m1.1.7.cmml" id="p1.1.m1.1.7" mathvariant="normal">&</mi>
- <mo xref="p1.1.m1.1.17.3.1.cmml" id="p1.1.m1.1.17.3.1a">â¢</mo>
- <mi xref="p1.1.m1.1.8.cmml" id="p1.1.m1.1.8">l</mi>
- <mo xref="p1.1.m1.1.17.3.1.cmml" id="p1.1.m1.1.17.3.1b">â¢</mo>
- <mi xref="p1.1.m1.1.9.cmml" id="p1.1.m1.1.9">t</mi>
- </mrow>
- <mo id="p1.1.m1.1.10">;</mo>
- <mrow xref="p1.1.m1.1.17.1.cmml" id="p1.1.m1.1.17.4">
- <mi xref="p1.1.m1.1.11.cmml" id="p1.1.m1.1.11">c</mi>
- <mo xref="p1.1.m1.1.17.4.1.cmml" id="p1.1.m1.1.17.4.1">â¢</mo>
- <mi xref="p1.1.m1.1.12.cmml" id="p1.1.m1.1.12" mathvariant="normal">&</mi>
- <mo xref="p1.1.m1.1.17.4.1.cmml" id="p1.1.m1.1.17.4.1a">â¢</mo>
- <mi xref="p1.1.m1.1.13.cmml" id="p1.1.m1.1.13">l</mi>
- <mo xref="p1.1.m1.1.17.4.1.cmml" id="p1.1.m1.1.17.4.1b">â¢</mo>
- <mi xref="p1.1.m1.1.14.cmml" id="p1.1.m1.1.14">t</mi>
- </mrow>
- <mo id="p1.1.m1.1.15">;</mo>
- <mi xref="p1.1.m1.1.16.cmml" id="p1.1.m1.1.16">d</mi>
- </mrow>
- <annotation-xml id="p1.1.m1.1b" encoding="MathML-Content">
- <list xref="p1.1.m1.1.17" id="p1.1.m1.1.17.1.cmml">
- <apply xref="p1.1.m1.1.17" id="p1.1.m1.1.17.2.cmml">
- <times xref="p1.1.m1.1.17.2.1" id="p1.1.m1.1.17.2.1.cmml"/>
- <ci xref="p1.1.m1.1.1" id="p1.1.m1.1.1.cmml">ð‘Ž</ci>
- <ci xref="p1.1.m1.1.2" id="p1.1.m1.1.2.cmml">&</ci>
- <ci xref="p1.1.m1.1.3" id="p1.1.m1.1.3.cmml">ð‘™</ci>
- <ci xref="p1.1.m1.1.4" id="p1.1.m1.1.4.cmml">ð‘¡</ci>
- </apply>
- <apply xref="p1.1.m1.1.17" id="p1.1.m1.1.17.3.cmml">
- <times xref="p1.1.m1.1.17.3.1" id="p1.1.m1.1.17.3.1.cmml"/>
- <ci xref="p1.1.m1.1.6" id="p1.1.m1.1.6.cmml">ð‘</ci>
- <ci xref="p1.1.m1.1.7" id="p1.1.m1.1.7.cmml">&</ci>
- <ci xref="p1.1.m1.1.8" id="p1.1.m1.1.8.cmml">ð‘™</ci>
- <ci xref="p1.1.m1.1.9" id="p1.1.m1.1.9.cmml">ð‘¡</ci>
- </apply>
- <apply xref="p1.1.m1.1.17" id="p1.1.m1.1.17.4.cmml">
- <times xref="p1.1.m1.1.17.4.1" id="p1.1.m1.1.17.4.1.cmml"/>
- <ci xref="p1.1.m1.1.11" id="p1.1.m1.1.11.cmml">ð‘</ci>
- <ci xref="p1.1.m1.1.12" id="p1.1.m1.1.12.cmml">&</ci>
- <ci xref="p1.1.m1.1.13" id="p1.1.m1.1.13.cmml">ð‘™</ci>
- <ci xref="p1.1.m1.1.14" id="p1.1.m1.1.14.cmml">ð‘¡</ci>
- </apply>
- <ci xref="p1.1.m1.1.16" id="p1.1.m1.1.16.cmml">ð‘‘</ci>
- </list>
- </annotation-xml>
- <annotation id="p1.1.m1.1c" encoding="application/x-tex">a&lt;b&lt;c&lt;d</annotation>
- </semantics>
-</math> and <math id="-7049119032560972238" display="inline" alttext="e&lt;f" class="ltx_Math" xmlns="http://www.w3.org/1998/Math/MathML">
- <semantics id="p1.1.m1.1a">
- <mrow xref="p1.1.m1.1.7.1.cmml" id="p1.1.m1.1.7">
- <mrow xref="p1.1.m1.1.7.1.cmml" id="p1.1.m1.1.7.2">
- <mi xref="p1.1.m1.1.1.cmml" id="p1.1.m1.1.1">e</mi>
- <mo xref="p1.1.m1.1.7.2.1.cmml" id="p1.1.m1.1.7.2.1">â¢</mo>
- <mi xref="p1.1.m1.1.2.cmml" id="p1.1.m1.1.2" mathvariant="normal">&</mi>
- <mo xref="p1.1.m1.1.7.2.1.cmml" id="p1.1.m1.1.7.2.1a">â¢</mo>
- <mi xref="p1.1.m1.1.3.cmml" id="p1.1.m1.1.3">l</mi>
- <mo xref="p1.1.m1.1.7.2.1.cmml" id="p1.1.m1.1.7.2.1b">â¢</mo>
- <mi xref="p1.1.m1.1.4.cmml" id="p1.1.m1.1.4">t</mi>
- </mrow>
- <mo id="p1.1.m1.1.5">;</mo>
- <mi xref="p1.1.m1.1.6.cmml" id="p1.1.m1.1.6">f</mi>
- </mrow>
- <annotation-xml id="p1.1.m1.1b" encoding="MathML-Content">
- <list xref="p1.1.m1.1.7" id="p1.1.m1.1.7.1.cmml">
- <apply xref="p1.1.m1.1.7" id="p1.1.m1.1.7.2.cmml">
- <times xref="p1.1.m1.1.7.2.1" id="p1.1.m1.1.7.2.1.cmml"/>
- <ci xref="p1.1.m1.1.1" id="p1.1.m1.1.1.cmml">ð‘’</ci>
- <ci xref="p1.1.m1.1.2" id="p1.1.m1.1.2.cmml">&</ci>
- <ci xref="p1.1.m1.1.3" id="p1.1.m1.1.3.cmml">ð‘™</ci>
- <ci xref="p1.1.m1.1.4" id="p1.1.m1.1.4.cmml">ð‘¡</ci>
- </apply>
- <ci xref="p1.1.m1.1.6" id="p1.1.m1.1.6.cmml">ð‘“</ci>
- </list>
- </annotation-xml>
- <annotation id="p1.1.m1.1c" encoding="application/x-tex">e&lt;f</annotation>
- </semantics>
-</math>.</p><p xmlns="http://www.w3.org/1999/xhtml">Using the <code class="code">Mode</code> attribute of a <code class="code">Display</code> element formulae like</p><p class="center" xmlns="http://www.w3.org/1999/xhtml">\[a \longrightarrow a \bmod m\prime\]</p><p xmlns="http://www.w3.org/1999/xhtml">can also be displayed nicely in text and HTML output.</p> </body>
+ <body> <h4 xmlns="http://www.w3.org/1999/xhtml">2.3 <span class="Heading">Crossreferencing</span></h4><p xmlns="http://www.w3.org/1999/xhtml">[→ <a shape="rect" href="chapB_mj.html#X7D19CF4782309661"><span class="RefLink">B.8</span></a>]</p><p xmlns="http://www.w3.org/1999/xhtml">In this section we demonstrate various references to parts of this document. Here is a reference to this section: <a shape="rect" href="chap2_mj.html#X833C410D85CF96A4"><span class="RefLink">2.3</span></a>. Here is a reference to chapter <a shape="rect" href="chap1_mj.html#X80E2AD7481DD69D9"><span class="RefLink">1</span></a>, to appendix <a shape="rect" href="chapA_mj.html#X7B53252784137533"><span class="RefLink">A</span></a>, and to subsection <a shape="rect" href="chap1_mj.html#X7E193BD379F58A4C"><span class="RefLink">1.1-1</span></a>.</p><p xmlns="http://www.w3.org/1999/xhtml">We distinguish among others references to functions (see <code class="func">f</code> (<a shape="rect" href="chap1_mj.html#X7FA1D0937FA1D093"><span class="RefLink">1.2-1</span></a>)), to methods with tricky name (see <code class="func">\^\{\}\[\]\<\&</code> (<a shape="rect" href="chap1_mj.html#X822B5C487B29E799"><span class="RefLink">1.2-2</span></a>)), to operations (see <code class="func">MyOperation</code> (<a shape="rect" href="chap1_mj.html#X7D33C2597988F481"><span class="RefLink">1.2-3</span></a>)), to methods (see <code class="func">MyOperation</code> (<a shape="rect" href="chap1_mj.html#X783DCD4E826289D4"><span class="RefLink">1.2-4</span></a>) or <code class="func">MyOperation</code> (<a shape="rect" href="chap1_mj.html#X7A5F4A287D06988C"><span class="RefLink">1.2-5</span></a>)), to filters (see <code class="func">IsBla</code> (<a shape="rect" href="chap1_mj.html#X82954B687D2DF3C2"><span class="RefLink">1.2-6</span></a>)), to properties (see <code class="func">IsBlubb</code> (<a shape="rect" href="chap1_mj.html#X80C364DD7C919CCE"><span class="RefLink">1.2-7</span></a>)), to attributes (see <code class="func">NumberBlobbs</code> (<a shape="rect" href="chap1_mj.html#X8052A45E7F9F054C"><span class="RefLink">1.2-8</span></a>)), to variables (<code class="func">AllBlibbs</code> (<a shape="rect" href="chap1_mj.html#X7C00E05A7DDEF003"><span class="RefLink">1.2-9</span></a>)), to families (see <code class="func">BlibbsFamily</code> (<a shape="rect" href="chap1_mj.html#X7CBC935A8142E374"><span class="RefLink">1.2-10</span></a>)), and to info classes (see <code class="func">InfoBlibbs</code> (<a shape="rect" href="chap1_mj.html#X84D7D77378AD030A"><span class="RefLink">1.2-11</span></a>)).</p><p xmlns="http://www.w3.org/1999/xhtml">There are also references to labels: see <a shape="rect" href="chap2_mj.html#X833C410D85CF96A4"><span class="RefLink">here</span></a>, to other books: see <a shape="rect" href="../../pkg/GAPDoc-1.5.1/doc/chap3_mj.html#X7B76F6F786521F6B"><span class="RefLink">GAPDoc: What is a DTD?</span></a> or <code class="func">IsSubgroup</code> (<a shape="rect" href="../../../doc/ref/chap40_mj.html#X7839D8927E778334"><span class="RefLink">Reference: IsSubgroup</span></a>) in the <strong class="pkg">GAP</strong> reference manual.</p><p xmlns="http://www.w3.org/1999/xhtml">References to sections come in two styles: <a shape="rect" href="chap1_mj.html#X80E2AD7481DD69D9"><span class="RefLink">1</span></a> or <a shape="rect" href="chap1_mj.html#X80E2AD7481DD69D9"><span class="RefLink"><span class="Heading">Sectioning Elements</span></span></a>.</p><p xmlns="http://www.w3.org/1999/xhtml">Another type of cross referencing is bibliography. Here is a citation: <a shape="rect" href="chapBib_mj.html#biBCR1">[CR81, (5.22)]</a> is an interesting lemma.</p><p xmlns="http://www.w3.org/1999/xhtml">There are also URLs:</p><p xmlns="http://www.w3.org/1999/xhtml"><span class="URL"><a shape="rect" href="http://www.math.rwth-aachen.de/">http://www.math.rwth-aachen.de/</a></span></p><p xmlns="http://www.w3.org/1999/xhtml">Email addresses have a special element: <span class="URL"><a shape="rect" href="mailto:Frank.Luebeck@Math.RWTH-Aachen.De">Frank.Luebeck@Math.RWTH-Aachen.De</a></span></p><p xmlns="http://www.w3.org/1999/xhtml">and Homepages another one: <span class="URL"><a shape="rect" href="http://www-groups.mcs.st-and.ac.uk/~neunhoef/">http://www-groups.mcs.st-and.ac.uk/~neunhoef/</a></span></p><p xmlns="http://www.w3.org/1999/xhtml">And here is a link to the <span class="URL"><a shape="rect" href="http://www.math.rwth-aachen.de/~Frank.Luebeck/gap/EDIM/index.html#ARCHS"><strong class="pkg">EDIM</strong> archives</a></span>.</p><p xmlns="http://www.w3.org/1999/xhtml">One can generate index entries as follows (look up the words "TeX-UserGroup", "RWTH", "Aachen, Hauptbahnhof", and "<strong class="pkg">GAP</strong>, <strong class="pkg">GAPDoc</strong>").</p> </body>
</html>
\ No newline at end of file
diff --git a/export/mwscrawler/4.html b/export/mwscrawler/4.html
index ea36980..296d9fc 100644
--- a/export/mwscrawler/4.html
+++ b/export/mwscrawler/4.html
@@ -1,51 +1,301 @@
<html>
<head>
- <title>TODO</title>
- <meta name="url" content="http://www.gap-system.org/Manuals/doc/ref/TODO#X7A480B9A795EF264"></meta>
+ <title>2.2 Formulae</title>
+ <meta name="url" content="http://www.gap-system.org/Manuals/doc/ref/TODO#X7AA5BF0279938BE0"></meta>
</head>
- <body> <h4 xmlns="http://www.w3.org/1999/xhtml">2.1 <span class="Heading">Various types of text</span></h4><p xmlns="http://www.w3.org/1999/xhtml">[→ <a shape="rect" href="chapB_mj.html#X82A731CA83FB9ADD"><span class="RefLink">B.5</span></a>]</p><p xmlns="http://www.w3.org/1999/xhtml">In this section we present examples for all the various types of text that are possible in <strong class="pkg">GAPDoc</strong>:</p><ul xmlns="http://www.w3.org/1999/xhtml"><li><p><em>This</em> is <em>emphasized</em>.</p>
+ <body> <h4 xmlns="http://www.w3.org/1999/xhtml">2.2 <span class="Heading">Formulae</span></h4><p xmlns="http://www.w3.org/1999/xhtml">[→ <a shape="rect" href="chapB_mj.html#X8516FAA07A95BBB5"><span class="RefLink">B.7</span></a>]</p><p xmlns="http://www.w3.org/1999/xhtml">There are three types of formulae.</p><p xmlns="http://www.w3.org/1999/xhtml">The first is the <em>normal math mode</em> of LaTeX: <math id="-1712973365669085523" display="inline" alttext="b_{i}\cdot b_{j}=\sum_{k=1}^{d}h_{ijk}b_{k}" class="ltx_Math" xmlns="http://www.w3.org/1998/Math/MathML">
+ <semantics id="p1.1.m1.1a">
+ <mrow xref="p1.1.m1.1.14.cmml" id="p1.1.m1.1.14">
+ <mrow xref="p1.1.m1.1.14.1.cmml" id="p1.1.m1.1.14.1">
+ <msub xref="p1.1.m1.1.14.1.1.cmml" id="p1.1.m1.1.14.1.1">
+ <mi xref="p1.1.m1.1.1.cmml" id="p1.1.m1.1.1">b</mi>
+ <mi xref="p1.1.m1.1.2.1.cmml" id="p1.1.m1.1.2.1">i</mi>
+ </msub>
+ <mo xref="p1.1.m1.1.3.cmml" id="p1.1.m1.1.3">â‹…</mo>
+ <msub xref="p1.1.m1.1.14.1.2.cmml" id="p1.1.m1.1.14.1.2">
+ <mi xref="p1.1.m1.1.4.cmml" id="p1.1.m1.1.4">b</mi>
+ <mi xref="p1.1.m1.1.5.1.cmml" id="p1.1.m1.1.5.1">j</mi>
+ </msub>
+ </mrow>
+ <mo xref="p1.1.m1.1.6.cmml" id="p1.1.m1.1.6">=</mo>
+ <mrow xref="p1.1.m1.1.14.2.cmml" id="p1.1.m1.1.14.2">
+ <msubsup xref="p1.1.m1.1.14.2.1.cmml" id="p1.1.m1.1.14.2.1">
+ <mo xref="p1.1.m1.1.7.cmml" id="p1.1.m1.1.7" symmetric="true" largeop="true">∑</mo>
+ <mrow xref="p1.1.m1.1.8.1.cmml" id="p1.1.m1.1.8.1">
+ <mi xref="p1.1.m1.1.8.1.1.cmml" id="p1.1.m1.1.8.1.1">k</mi>
+ <mo xref="p1.1.m1.1.8.1.2.cmml" id="p1.1.m1.1.8.1.2">=</mo>
+ <mn xref="p1.1.m1.1.8.1.3.cmml" id="p1.1.m1.1.8.1.3">1</mn>
+ </mrow>
+ <mi xref="p1.1.m1.1.9.1.cmml" id="p1.1.m1.1.9.1">d</mi>
+ </msubsup>
+ <mrow xref="p1.1.m1.1.14.2.2.cmml" id="p1.1.m1.1.14.2.2">
+ <msub xref="p1.1.m1.1.14.2.2.2.cmml" id="p1.1.m1.1.14.2.2.2">
+ <mi xref="p1.1.m1.1.10.cmml" id="p1.1.m1.1.10">h</mi>
+ <mrow xref="p1.1.m1.1.11.1.cmml" id="p1.1.m1.1.11.1">
+ <mi xref="p1.1.m1.1.11.1.1.cmml" id="p1.1.m1.1.11.1.1">i</mi>
+ <mo xref="p1.1.m1.1.11.1.4.cmml" id="p1.1.m1.1.11.1.4">â¢</mo>
+ <mi xref="p1.1.m1.1.11.1.2.cmml" id="p1.1.m1.1.11.1.2">j</mi>
+ <mo xref="p1.1.m1.1.11.1.4.cmml" id="p1.1.m1.1.11.1.4a">â¢</mo>
+ <mi xref="p1.1.m1.1.11.1.3.cmml" id="p1.1.m1.1.11.1.3">k</mi>
+ </mrow>
+ </msub>
+ <mo xref="p1.1.m1.1.14.2.2.1.cmml" id="p1.1.m1.1.14.2.2.1">â¢</mo>
+ <msub xref="p1.1.m1.1.14.2.2.3.cmml" id="p1.1.m1.1.14.2.2.3">
+ <mi xref="p1.1.m1.1.12.cmml" id="p1.1.m1.1.12">b</mi>
+ <mi xref="p1.1.m1.1.13.1.cmml" id="p1.1.m1.1.13.1">k</mi>
+ </msub>
+ </mrow>
+ </mrow>
+ </mrow>
+ <annotation-xml id="p1.1.m1.1b" encoding="MathML-Content">
+ <apply xref="p1.1.m1.1.14" id="p1.1.m1.1.14.cmml">
+ <eq xref="p1.1.m1.1.6" id="p1.1.m1.1.6.cmml"/>
+ <apply xref="p1.1.m1.1.14.1" id="p1.1.m1.1.14.1.cmml">
+ <ci xref="p1.1.m1.1.3" id="p1.1.m1.1.3.cmml">â‹…</ci>
+ <apply xref="p1.1.m1.1.14.1.1" id="p1.1.m1.1.14.1.1.cmml">
+ <csymbol id="p1.1.m1.1.14.1.1.1.cmml" cd="ambiguous">subscript</csymbol>
+ <ci xref="p1.1.m1.1.1" id="p1.1.m1.1.1.cmml">ð‘</ci>
+ <ci xref="p1.1.m1.1.2.1" id="p1.1.m1.1.2.1.cmml">ð‘–</ci>
+ </apply>
+ <apply xref="p1.1.m1.1.14.1.2" id="p1.1.m1.1.14.1.2.cmml">
+ <csymbol id="p1.1.m1.1.14.1.2.1.cmml" cd="ambiguous">subscript</csymbol>
+ <ci xref="p1.1.m1.1.4" id="p1.1.m1.1.4.cmml">ð‘</ci>
+ <ci xref="p1.1.m1.1.5.1" id="p1.1.m1.1.5.1.cmml">ð‘—</ci>
+ </apply>
+ </apply>
+ <apply xref="p1.1.m1.1.14.2" id="p1.1.m1.1.14.2.cmml">
+ <apply xref="p1.1.m1.1.14.2.1" id="p1.1.m1.1.14.2.1.cmml">
+ <csymbol id="p1.1.m1.1.14.2.1.1.cmml" cd="ambiguous">superscript</csymbol>
+ <apply id="p1.1.m1.1.14.2.1.2.cmml">
+ <csymbol id="p1.1.m1.1.14.2.1.2.1.cmml" cd="ambiguous">subscript</csymbol>
+ <sum xref="p1.1.m1.1.7" id="p1.1.m1.1.7.cmml"/>
+ <apply xref="p1.1.m1.1.8.1" id="p1.1.m1.1.8.1.cmml">
+ <eq xref="p1.1.m1.1.8.1.2" id="p1.1.m1.1.8.1.2.cmml"/>
+ <ci xref="p1.1.m1.1.8.1.1" id="p1.1.m1.1.8.1.1.cmml">ð‘˜</ci>
+ <cn xref="p1.1.m1.1.8.1.3" id="p1.1.m1.1.8.1.3.cmml" type="integer">1</cn>
+ </apply>
+ </apply>
+ <ci xref="p1.1.m1.1.9.1" id="p1.1.m1.1.9.1.cmml">ð‘‘</ci>
+ </apply>
+ <apply xref="p1.1.m1.1.14.2.2" id="p1.1.m1.1.14.2.2.cmml">
+ <times xref="p1.1.m1.1.14.2.2.1" id="p1.1.m1.1.14.2.2.1.cmml"/>
+ <apply xref="p1.1.m1.1.14.2.2.2" id="p1.1.m1.1.14.2.2.2.cmml">
+ <csymbol id="p1.1.m1.1.14.2.2.2.1.cmml" cd="ambiguous">subscript</csymbol>
+ <ci xref="p1.1.m1.1.10" id="p1.1.m1.1.10.cmml">â„Ž</ci>
+ <apply xref="p1.1.m1.1.11.1" id="p1.1.m1.1.11.1.cmml">
+ <times xref="p1.1.m1.1.11.1.4" id="p1.1.m1.1.11.1.4.cmml"/>
+ <ci xref="p1.1.m1.1.11.1.1" id="p1.1.m1.1.11.1.1.cmml">ð‘–</ci>
+ <ci xref="p1.1.m1.1.11.1.2" id="p1.1.m1.1.11.1.2.cmml">ð‘—</ci>
+ <ci xref="p1.1.m1.1.11.1.3" id="p1.1.m1.1.11.1.3.cmml">ð‘˜</ci>
+ </apply>
+ </apply>
+ <apply xref="p1.1.m1.1.14.2.2.3" id="p1.1.m1.1.14.2.2.3.cmml">
+ <csymbol id="p1.1.m1.1.14.2.2.3.1.cmml" cd="ambiguous">subscript</csymbol>
+ <ci xref="p1.1.m1.1.12" id="p1.1.m1.1.12.cmml">ð‘</ci>
+ <ci xref="p1.1.m1.1.13.1" id="p1.1.m1.1.13.1.cmml">ð‘˜</ci>
+ </apply>
+ </apply>
+ </apply>
+ </apply>
+ </annotation-xml>
+ <annotation id="p1.1.m1.1c" encoding="application/x-tex">b_{i}\cdot b_{j}=\sum_{k=1}^{d}h_{ijk}b_{k}</annotation>
+ </semantics>
+</math>. Then there are <em>displayed formulae</em>:</p><p class="center" xmlns="http://www.w3.org/1999/xhtml">\[
+ \Longrightarrow \quad \left(\sum_{i=1}^d x_i b_i \right) \cdot
+ \left(\sum_{j=1}^d y_j b_j \right) =
+ \sum_{k=1}^d \left( \sum_{i,j} x_i y_j h_{ijk} \right) b_k
+\]</p><p xmlns="http://www.w3.org/1999/xhtml">If possible, use the <code class="code">Alt</code> element to specify a better readable text version of such a formula as in the following example:</p><pre xml:space="preserve" class="normal" xmlns="http://www.w3.org/1999/xhtml">
-</li><li><p><em>Keywords</em> are typeset like <code class="keyw">this</code> and <code class="keyw">that</code>.</p>
+ d d d
+ ----- ----- ----- -----
+ \ \ \ \
+ ==> ( ) x_i b_i )( ) y_j b_j ) = ) ( ) x_i y_j h_ijk ) b_k
+ / / / /
+ ----- ----- ----- -----
+ i = 1 j = 1 k = 1 i,j
-</li><li><p><em>Arguments</em> of functions have an element. They look like this: <var class="Arg">x</var> and <var class="Arg">y</var>.</p>
-
-</li><li><p><em>Code</em> can be written with the Code element: <code class="code">if x = y then Print("Equal"); fi;</code> or <code class="code">while true do Print("Hello"); od;</code>.</p>
-
-</li><li><p><em>Filenames</em> have their own element: <code class="file">/usr/local/ca/gap4r2</code> or <code class="file">pkg/xgap/doc</code>.</p>
-
-</li><li><p><em>Buttons</em>, <em>menus</em>, <em>menu entries</em>, and such things are also supported: <strong class="button">OK</strong> or <strong class="button">Cancel</strong>.</p>
-
-</li><li><p><em>Packages</em> are typeset like this: <strong class="pkg">Small Groups Library</strong></p>
-
-</li><li><p><em>Quoted</em> text: "This is a text in quotes."</p>
-
-</li></ul><p xmlns="http://www.w3.org/1999/xhtml"><em>Paragraphs</em> are separated by the empty <code class="code">Par</code> or <code class="code">P</code> element.</p><p xmlns="http://www.w3.org/1999/xhtml"><em>Alternatives</em> for different output formats: This is other than LaTeX output, namely: <b>HTML</b> output.</p><p xmlns="http://www.w3.org/1999/xhtml">There are also three elements to typeset "verbatim-like" text. (→ <a shape="rect" href="chapB_mj.html#X824BD70087820CF0"><span class="RefLink">B.6</span></a>)</p><p xmlns="http://www.w3.org/1999/xhtml">The first is a <em>Listing</em>:</p><div class="example" xmlns="http://www.w3.org/1999/xhtml"><pre xml:space="preserve">
-Sieve := function(n)
- # Returns the primes less than n
- local l,p,i;
- l := [1..n]; Unbind(l[1]);
- p := 2;
- while p^2 <= n do
- if IsBound(l[p]) then
- i := 2 * p;
- while i <= n do Unbind(l[i]); i := i + p; od;
- fi;
- p := p + 1;
- od;
- return Compacted(l);
-end;
-</pre></div><p xmlns="http://www.w3.org/1999/xhtml">Here is a <em>Log</em> of a <strong class="pkg">GAP</strong> session using this function:</p><div class="example" xmlns="http://www.w3.org/1999/xhtml"><pre xml:space="preserve">
-<span class="GAPprompt">gap></span> <span class="GAPinput">Sieve(100);</span>
-[ 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61,
- 67, 71, 73, 79, 83, 89, 97 ]
-<span class="GAPprompt">gap></span> <span class="GAPinput">Length(last);</span>
-25
-</pre></div><p xmlns="http://www.w3.org/1999/xhtml">Here is a <strong class="pkg">GAP</strong> <em>Example</em> session that is automatically tested:</p><div class="example" xmlns="http://www.w3.org/1999/xhtml"><pre xml:space="preserve">
-<span class="GAPprompt">gap></span> <span class="GAPinput">s := Size(CharacterTable("M"));</span>
-808017424794512875886459904961710757005754368000000000
-<span class="GAPprompt">gap></span> <span class="GAPinput">s < 10^53; </span>
-false
-<span class="GAPprompt">gap></span> <span class="GAPinput">s < 10^54;</span>
-true
-</pre></div> </body>
+</pre><p xmlns="http://www.w3.org/1999/xhtml">For small formulae without "difficult" parts use the <code class="code">M</code> element: <math id="7746720724519183758" display="inline" alttext="b_{i}" class="ltx_Math" xmlns="http://www.w3.org/1998/Math/MathML">
+ <semantics id="p1.1.m1.1a">
+ <msub xref="p1.1.m1.1.3.cmml" id="p1.1.m1.1.3">
+ <mi xref="p1.1.m1.1.1.cmml" id="p1.1.m1.1.1">b</mi>
+ <mi xref="p1.1.m1.1.2.1.cmml" id="p1.1.m1.1.2.1">i</mi>
+ </msub>
+ <annotation-xml id="p1.1.m1.1b" encoding="MathML-Content">
+ <apply xref="p1.1.m1.1.3" id="p1.1.m1.1.3.cmml">
+ <csymbol id="p1.1.m1.1.3.1.cmml" cd="ambiguous">subscript</csymbol>
+ <ci xref="p1.1.m1.1.1" id="p1.1.m1.1.1.cmml">ð‘</ci>
+ <ci xref="p1.1.m1.1.2.1" id="p1.1.m1.1.2.1.cmml">ð‘–</ci>
+ </apply>
+ </annotation-xml>
+ <annotation id="p1.1.m1.1c" encoding="application/x-tex">b_{i}</annotation>
+ </semantics>
+</math>, <math id="6850696246020381825" display="inline" alttext="x^{2}" class="ltx_Math" xmlns="http://www.w3.org/1998/Math/MathML">
+ <semantics id="p1.1.m1.1a">
+ <msup xref="p1.1.m1.1.3.cmml" id="p1.1.m1.1.3">
+ <mi xref="p1.1.m1.1.1.cmml" id="p1.1.m1.1.1">x</mi>
+ <mn xref="p1.1.m1.1.2.1.cmml" id="p1.1.m1.1.2.1">2</mn>
+ </msup>
+ <annotation-xml id="p1.1.m1.1b" encoding="MathML-Content">
+ <apply xref="p1.1.m1.1.3" id="p1.1.m1.1.3.cmml">
+ <csymbol id="p1.1.m1.1.3.1.cmml" cd="ambiguous">superscript</csymbol>
+ <ci xref="p1.1.m1.1.1" id="p1.1.m1.1.1.cmml">ð‘¥</ci>
+ <cn xref="p1.1.m1.1.2.1" id="p1.1.m1.1.2.1.cmml" type="integer">2</cn>
+ </apply>
+ </annotation-xml>
+ <annotation id="p1.1.m1.1c" encoding="application/x-tex">x^{2}</annotation>
+ </semantics>
+</math>, <math id="2377982914200794544" display="inline" alttext="x^{2}+2x+1=(x+1)^{2}" class="ltx_Math" xmlns="http://www.w3.org/1998/Math/MathML">
+ <semantics id="p1.1.m1.1a">
+ <mrow xref="p1.1.m1.1.15.cmml" id="p1.1.m1.1.15">
+ <mrow xref="p1.1.m1.1.15.1.cmml" id="p1.1.m1.1.15.1">
+ <msup xref="p1.1.m1.1.15.1.1.cmml" id="p1.1.m1.1.15.1.1">
+ <mi xref="p1.1.m1.1.1.cmml" id="p1.1.m1.1.1">x</mi>
+ <mn xref="p1.1.m1.1.2.1.cmml" id="p1.1.m1.1.2.1">2</mn>
+ </msup>
+ <mo xref="p1.1.m1.1.3.cmml" id="p1.1.m1.1.3">+</mo>
+ <mrow xref="p1.1.m1.1.15.1.2.cmml" id="p1.1.m1.1.15.1.2">
+ <mn xref="p1.1.m1.1.4.cmml" id="p1.1.m1.1.4">2</mn>
+ <mo xref="p1.1.m1.1.15.1.2.1.cmml" id="p1.1.m1.1.15.1.2.1">â¢</mo>
+ <mi xref="p1.1.m1.1.5.cmml" id="p1.1.m1.1.5">x</mi>
+ </mrow>
+ <mo xref="p1.1.m1.1.3.cmml" id="p1.1.m1.1.3a">+</mo>
+ <mn xref="p1.1.m1.1.7.cmml" id="p1.1.m1.1.7">1</mn>
+ </mrow>
+ <mo xref="p1.1.m1.1.8.cmml" id="p1.1.m1.1.8">=</mo>
+ <msup xref="p1.1.m1.1.15.2.cmml" id="p1.1.m1.1.15.2">
+ <mrow xref="p1.1.m1.1.15.2.2.2.cmml" id="p1.1.m1.1.15.2.2">
+ <mo id="p1.1.m1.1.9" stretchy="false">(</mo>
+ <mrow xref="p1.1.m1.1.15.2.2.2.cmml" id="p1.1.m1.1.15.2.2.2">
+ <mi xref="p1.1.m1.1.10.cmml" id="p1.1.m1.1.10">x</mi>
+ <mo xref="p1.1.m1.1.11.cmml" id="p1.1.m1.1.11">+</mo>
+ <mn xref="p1.1.m1.1.12.cmml" id="p1.1.m1.1.12">1</mn>
+ </mrow>
+ <mo id="p1.1.m1.1.13" stretchy="false">)</mo>
+ </mrow>
+ <mn xref="p1.1.m1.1.14.1.cmml" id="p1.1.m1.1.14.1">2</mn>
+ </msup>
+ </mrow>
+ <annotation-xml id="p1.1.m1.1b" encoding="MathML-Content">
+ <apply xref="p1.1.m1.1.15" id="p1.1.m1.1.15.cmml">
+ <eq xref="p1.1.m1.1.8" id="p1.1.m1.1.8.cmml"/>
+ <apply xref="p1.1.m1.1.15.1" id="p1.1.m1.1.15.1.cmml">
+ <plus xref="p1.1.m1.1.3" id="p1.1.m1.1.3.cmml"/>
+ <apply xref="p1.1.m1.1.15.1.1" id="p1.1.m1.1.15.1.1.cmml">
+ <csymbol id="p1.1.m1.1.15.1.1.1.cmml" cd="ambiguous">superscript</csymbol>
+ <ci xref="p1.1.m1.1.1" id="p1.1.m1.1.1.cmml">ð‘¥</ci>
+ <cn xref="p1.1.m1.1.2.1" id="p1.1.m1.1.2.1.cmml" type="integer">2</cn>
+ </apply>
+ <apply xref="p1.1.m1.1.15.1.2" id="p1.1.m1.1.15.1.2.cmml">
+ <times xref="p1.1.m1.1.15.1.2.1" id="p1.1.m1.1.15.1.2.1.cmml"/>
+ <cn xref="p1.1.m1.1.4" id="p1.1.m1.1.4.cmml" type="integer">2</cn>
+ <ci xref="p1.1.m1.1.5" id="p1.1.m1.1.5.cmml">ð‘¥</ci>
+ </apply>
+ <cn xref="p1.1.m1.1.7" id="p1.1.m1.1.7.cmml" type="integer">1</cn>
+ </apply>
+ <apply xref="p1.1.m1.1.15.2" id="p1.1.m1.1.15.2.cmml">
+ <csymbol id="p1.1.m1.1.15.2.1.cmml" cd="ambiguous">superscript</csymbol>
+ <apply xref="p1.1.m1.1.15.2.2" id="p1.1.m1.1.15.2.2.2.cmml">
+ <plus xref="p1.1.m1.1.11" id="p1.1.m1.1.11.cmml"/>
+ <ci xref="p1.1.m1.1.10" id="p1.1.m1.1.10.cmml">ð‘¥</ci>
+ <cn xref="p1.1.m1.1.12" id="p1.1.m1.1.12.cmml" type="integer">1</cn>
+ </apply>
+ <cn xref="p1.1.m1.1.14.1" id="p1.1.m1.1.14.1.cmml" type="integer">2</cn>
+ </apply>
+ </apply>
+ </annotation-xml>
+ <annotation id="p1.1.m1.1c" encoding="application/x-tex">x^{2}+2x+1=(x+1)^{2}</annotation>
+ </semantics>
+</math>. Note that here whitespace matters for text (or HTML) output.</p><p xmlns="http://www.w3.org/1999/xhtml">Here are two formulae containing less than characters which are special characters for XML: <math id="-6773294668414572344" display="inline" alttext="a&lt;b&lt;c&lt;d" class="ltx_Math" xmlns="http://www.w3.org/1998/Math/MathML">
+ <semantics id="p1.1.m1.1a">
+ <mrow xref="p1.1.m1.1.17.1.cmml" id="p1.1.m1.1.17">
+ <mrow xref="p1.1.m1.1.17.1.cmml" id="p1.1.m1.1.17.2">
+ <mi xref="p1.1.m1.1.1.cmml" id="p1.1.m1.1.1">a</mi>
+ <mo xref="p1.1.m1.1.17.2.1.cmml" id="p1.1.m1.1.17.2.1">â¢</mo>
+ <mi xref="p1.1.m1.1.2.cmml" id="p1.1.m1.1.2" mathvariant="normal">&</mi>
+ <mo xref="p1.1.m1.1.17.2.1.cmml" id="p1.1.m1.1.17.2.1a">â¢</mo>
+ <mi xref="p1.1.m1.1.3.cmml" id="p1.1.m1.1.3">l</mi>
+ <mo xref="p1.1.m1.1.17.2.1.cmml" id="p1.1.m1.1.17.2.1b">â¢</mo>
+ <mi xref="p1.1.m1.1.4.cmml" id="p1.1.m1.1.4">t</mi>
+ </mrow>
+ <mo id="p1.1.m1.1.5">;</mo>
+ <mrow xref="p1.1.m1.1.17.1.cmml" id="p1.1.m1.1.17.3">
+ <mi xref="p1.1.m1.1.6.cmml" id="p1.1.m1.1.6">b</mi>
+ <mo xref="p1.1.m1.1.17.3.1.cmml" id="p1.1.m1.1.17.3.1">â¢</mo>
+ <mi xref="p1.1.m1.1.7.cmml" id="p1.1.m1.1.7" mathvariant="normal">&</mi>
+ <mo xref="p1.1.m1.1.17.3.1.cmml" id="p1.1.m1.1.17.3.1a">â¢</mo>
+ <mi xref="p1.1.m1.1.8.cmml" id="p1.1.m1.1.8">l</mi>
+ <mo xref="p1.1.m1.1.17.3.1.cmml" id="p1.1.m1.1.17.3.1b">â¢</mo>
+ <mi xref="p1.1.m1.1.9.cmml" id="p1.1.m1.1.9">t</mi>
+ </mrow>
+ <mo id="p1.1.m1.1.10">;</mo>
+ <mrow xref="p1.1.m1.1.17.1.cmml" id="p1.1.m1.1.17.4">
+ <mi xref="p1.1.m1.1.11.cmml" id="p1.1.m1.1.11">c</mi>
+ <mo xref="p1.1.m1.1.17.4.1.cmml" id="p1.1.m1.1.17.4.1">â¢</mo>
+ <mi xref="p1.1.m1.1.12.cmml" id="p1.1.m1.1.12" mathvariant="normal">&</mi>
+ <mo xref="p1.1.m1.1.17.4.1.cmml" id="p1.1.m1.1.17.4.1a">â¢</mo>
+ <mi xref="p1.1.m1.1.13.cmml" id="p1.1.m1.1.13">l</mi>
+ <mo xref="p1.1.m1.1.17.4.1.cmml" id="p1.1.m1.1.17.4.1b">â¢</mo>
+ <mi xref="p1.1.m1.1.14.cmml" id="p1.1.m1.1.14">t</mi>
+ </mrow>
+ <mo id="p1.1.m1.1.15">;</mo>
+ <mi xref="p1.1.m1.1.16.cmml" id="p1.1.m1.1.16">d</mi>
+ </mrow>
+ <annotation-xml id="p1.1.m1.1b" encoding="MathML-Content">
+ <list xref="p1.1.m1.1.17" id="p1.1.m1.1.17.1.cmml">
+ <apply xref="p1.1.m1.1.17" id="p1.1.m1.1.17.2.cmml">
+ <times xref="p1.1.m1.1.17.2.1" id="p1.1.m1.1.17.2.1.cmml"/>
+ <ci xref="p1.1.m1.1.1" id="p1.1.m1.1.1.cmml">ð‘Ž</ci>
+ <ci xref="p1.1.m1.1.2" id="p1.1.m1.1.2.cmml">&</ci>
+ <ci xref="p1.1.m1.1.3" id="p1.1.m1.1.3.cmml">ð‘™</ci>
+ <ci xref="p1.1.m1.1.4" id="p1.1.m1.1.4.cmml">ð‘¡</ci>
+ </apply>
+ <apply xref="p1.1.m1.1.17" id="p1.1.m1.1.17.3.cmml">
+ <times xref="p1.1.m1.1.17.3.1" id="p1.1.m1.1.17.3.1.cmml"/>
+ <ci xref="p1.1.m1.1.6" id="p1.1.m1.1.6.cmml">ð‘</ci>
+ <ci xref="p1.1.m1.1.7" id="p1.1.m1.1.7.cmml">&</ci>
+ <ci xref="p1.1.m1.1.8" id="p1.1.m1.1.8.cmml">ð‘™</ci>
+ <ci xref="p1.1.m1.1.9" id="p1.1.m1.1.9.cmml">ð‘¡</ci>
+ </apply>
+ <apply xref="p1.1.m1.1.17" id="p1.1.m1.1.17.4.cmml">
+ <times xref="p1.1.m1.1.17.4.1" id="p1.1.m1.1.17.4.1.cmml"/>
+ <ci xref="p1.1.m1.1.11" id="p1.1.m1.1.11.cmml">ð‘</ci>
+ <ci xref="p1.1.m1.1.12" id="p1.1.m1.1.12.cmml">&</ci>
+ <ci xref="p1.1.m1.1.13" id="p1.1.m1.1.13.cmml">ð‘™</ci>
+ <ci xref="p1.1.m1.1.14" id="p1.1.m1.1.14.cmml">ð‘¡</ci>
+ </apply>
+ <ci xref="p1.1.m1.1.16" id="p1.1.m1.1.16.cmml">ð‘‘</ci>
+ </list>
+ </annotation-xml>
+ <annotation id="p1.1.m1.1c" encoding="application/x-tex">a&lt;b&lt;c&lt;d</annotation>
+ </semantics>
+</math> and <math id="-5879058293188165850" display="inline" alttext="e&lt;f" class="ltx_Math" xmlns="http://www.w3.org/1998/Math/MathML">
+ <semantics id="p1.1.m1.1a">
+ <mrow xref="p1.1.m1.1.7.1.cmml" id="p1.1.m1.1.7">
+ <mrow xref="p1.1.m1.1.7.1.cmml" id="p1.1.m1.1.7.2">
+ <mi xref="p1.1.m1.1.1.cmml" id="p1.1.m1.1.1">e</mi>
+ <mo xref="p1.1.m1.1.7.2.1.cmml" id="p1.1.m1.1.7.2.1">â¢</mo>
+ <mi xref="p1.1.m1.1.2.cmml" id="p1.1.m1.1.2" mathvariant="normal">&</mi>
+ <mo xref="p1.1.m1.1.7.2.1.cmml" id="p1.1.m1.1.7.2.1a">â¢</mo>
+ <mi xref="p1.1.m1.1.3.cmml" id="p1.1.m1.1.3">l</mi>
+ <mo xref="p1.1.m1.1.7.2.1.cmml" id="p1.1.m1.1.7.2.1b">â¢</mo>
+ <mi xref="p1.1.m1.1.4.cmml" id="p1.1.m1.1.4">t</mi>
+ </mrow>
+ <mo id="p1.1.m1.1.5">;</mo>
+ <mi xref="p1.1.m1.1.6.cmml" id="p1.1.m1.1.6">f</mi>
+ </mrow>
+ <annotation-xml id="p1.1.m1.1b" encoding="MathML-Content">
+ <list xref="p1.1.m1.1.7" id="p1.1.m1.1.7.1.cmml">
+ <apply xref="p1.1.m1.1.7" id="p1.1.m1.1.7.2.cmml">
+ <times xref="p1.1.m1.1.7.2.1" id="p1.1.m1.1.7.2.1.cmml"/>
+ <ci xref="p1.1.m1.1.1" id="p1.1.m1.1.1.cmml">ð‘’</ci>
+ <ci xref="p1.1.m1.1.2" id="p1.1.m1.1.2.cmml">&</ci>
+ <ci xref="p1.1.m1.1.3" id="p1.1.m1.1.3.cmml">ð‘™</ci>
+ <ci xref="p1.1.m1.1.4" id="p1.1.m1.1.4.cmml">ð‘¡</ci>
+ </apply>
+ <ci xref="p1.1.m1.1.6" id="p1.1.m1.1.6.cmml">ð‘“</ci>
+ </list>
+ </annotation-xml>
+ <annotation id="p1.1.m1.1c" encoding="application/x-tex">e&lt;f</annotation>
+ </semantics>
+</math>.</p><p xmlns="http://www.w3.org/1999/xhtml">Using the <code class="code">Mode</code> attribute of a <code class="code">Display</code> element formulae like</p><p class="center" xmlns="http://www.w3.org/1999/xhtml">\[a \longrightarrow a \bmod m\prime\]</p><p xmlns="http://www.w3.org/1999/xhtml">can also be displayed nicely in text and HTML output.</p> </body>
</html>
\ No newline at end of file
diff --git a/export/mwscrawler/5.html b/export/mwscrawler/5.html
index e2c34ab..379e010 100644
--- a/export/mwscrawler/5.html
+++ b/export/mwscrawler/5.html
@@ -1,23 +1,51 @@
<html>
<head>
- <title>TODO</title>
- <meta name="url" content="http://www.gap-system.org/Manuals/doc/ref/TODO#X82793A7E7A3F2A54"></meta>
+ <title>2.1 Various types of text</title>
+ <meta name="url" content="http://www.gap-system.org/Manuals/doc/ref/TODO#X7A480B9A795EF264"></meta>
</head>
- <body> <div class="ChapSects" xmlns="http://www.w3.org/1999/xhtml"><a shape="rect" href="chap2_mj.html#X82793A7E7A3F2A54">2 <span class="Heading">Other Markup</span></a>
-<div class="ContSect"><span class="tocline"><span class="nocss">Â </span><a shape="rect" href="chap2_mj.html#X7A480B9A795EF264">2.1 <span class="Heading">Various types of text</span></a>
-</span>
-</div>
-<div class="ContSect"><span class="tocline"><span class="nocss">Â </span><a shape="rect" href="chap2_mj.html#X7AA5BF0279938BE0">2.2 <span class="Heading">Formulae</span></a>
-</span>
-</div>
-<div class="ContSect"><span class="tocline"><span class="nocss">Â </span><a shape="rect" href="chap2_mj.html#X833C410D85CF96A4">2.3 <span class="Heading">Crossreferencing</span></a>
-</span>
-</div>
-<div class="ContSect"><span class="tocline"><span class="nocss">Â </span><a shape="rect" href="chap2_mj.html#X7F10E951789D6EDF">2.4 <span class="Heading">Lists and Tables</span></a>
-</span>
-</div>
-<div class="ContSect"><span class="tocline"><span class="nocss">Â </span><a shape="rect" href="chap2_mj.html#X83A355E68485D6D1">2.5 <span class="Heading">Entities and Special Characters</span></a>
-</span>
-</div>
-</div><h3 xmlns="http://www.w3.org/1999/xhtml">2 <span class="Heading">Other Markup</span></h3> </body>
+ <body> <h4 xmlns="http://www.w3.org/1999/xhtml">2.1 <span class="Heading">Various types of text</span></h4><p xmlns="http://www.w3.org/1999/xhtml">[→ <a shape="rect" href="chapB_mj.html#X82A731CA83FB9ADD"><span class="RefLink">B.5</span></a>]</p><p xmlns="http://www.w3.org/1999/xhtml">In this section we present examples for all the various types of text that are possible in <strong class="pkg">GAPDoc</strong>:</p><ul xmlns="http://www.w3.org/1999/xhtml"><li><p><em>This</em> is <em>emphasized</em>.</p>
+
+</li><li><p><em>Keywords</em> are typeset like <code class="keyw">this</code> and <code class="keyw">that</code>.</p>
+
+</li><li><p><em>Arguments</em> of functions have an element. They look like this: <var class="Arg">x</var> and <var class="Arg">y</var>.</p>
+
+</li><li><p><em>Code</em> can be written with the Code element: <code class="code">if x = y then Print("Equal"); fi;</code> or <code class="code">while true do Print("Hello"); od;</code>.</p>
+
+</li><li><p><em>Filenames</em> have their own element: <code class="file">/usr/local/ca/gap4r2</code> or <code class="file">pkg/xgap/doc</code>.</p>
+
+</li><li><p><em>Buttons</em>, <em>menus</em>, <em>menu entries</em>, and such things are also supported: <strong class="button">OK</strong> or <strong class="button">Cancel</strong>.</p>
+
+</li><li><p><em>Packages</em> are typeset like this: <strong class="pkg">Small Groups Library</strong></p>
+
+</li><li><p><em>Quoted</em> text: "This is a text in quotes."</p>
+
+</li></ul><p xmlns="http://www.w3.org/1999/xhtml"><em>Paragraphs</em> are separated by the empty <code class="code">Par</code> or <code class="code">P</code> element.</p><p xmlns="http://www.w3.org/1999/xhtml"><em>Alternatives</em> for different output formats: This is other than LaTeX output, namely: <b>HTML</b> output.</p><p xmlns="http://www.w3.org/1999/xhtml">There are also three elements to typeset "verbatim-like" text. (→ <a shape="rect" href="chapB_mj.html#X824BD70087820CF0"><span class="RefLink">B.6</span></a>)</p><p xmlns="http://www.w3.org/1999/xhtml">The first is a <em>Listing</em>:</p><div class="example" xmlns="http://www.w3.org/1999/xhtml"><pre xml:space="preserve">
+Sieve := function(n)
+ # Returns the primes less than n
+ local l,p,i;
+ l := [1..n]; Unbind(l[1]);
+ p := 2;
+ while p^2 <= n do
+ if IsBound(l[p]) then
+ i := 2 * p;
+ while i <= n do Unbind(l[i]); i := i + p; od;
+ fi;
+ p := p + 1;
+ od;
+ return Compacted(l);
+end;
+</pre></div><p xmlns="http://www.w3.org/1999/xhtml">Here is a <em>Log</em> of a <strong class="pkg">GAP</strong> session using this function:</p><div class="example" xmlns="http://www.w3.org/1999/xhtml"><pre xml:space="preserve">
+<span class="GAPprompt">gap></span> <span class="GAPinput">Sieve(100);</span>
+[ 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61,
+ 67, 71, 73, 79, 83, 89, 97 ]
+<span class="GAPprompt">gap></span> <span class="GAPinput">Length(last);</span>
+25
+</pre></div><p xmlns="http://www.w3.org/1999/xhtml">Here is a <strong class="pkg">GAP</strong> <em>Example</em> session that is automatically tested:</p><div class="example" xmlns="http://www.w3.org/1999/xhtml"><pre xml:space="preserve">
+<span class="GAPprompt">gap></span> <span class="GAPinput">s := Size(CharacterTable("M"));</span>
+808017424794512875886459904961710757005754368000000000
+<span class="GAPprompt">gap></span> <span class="GAPinput">s < 10^53; </span>
+false
+<span class="GAPprompt">gap></span> <span class="GAPinput">s < 10^54;</span>
+true
+</pre></div> </body>
</html>
\ No newline at end of file
diff --git a/export/mwscrawler/6.html b/export/mwscrawler/6.html
new file mode 100644
index 0000000..e2c34ab
--- /dev/null
+++ b/export/mwscrawler/6.html
@@ -0,0 +1,23 @@
+<html>
+ <head>
+ <title>TODO</title>
+ <meta name="url" content="http://www.gap-system.org/Manuals/doc/ref/TODO#X82793A7E7A3F2A54"></meta>
+ </head>
+ <body> <div class="ChapSects" xmlns="http://www.w3.org/1999/xhtml"><a shape="rect" href="chap2_mj.html#X82793A7E7A3F2A54">2 <span class="Heading">Other Markup</span></a>
+<div class="ContSect"><span class="tocline"><span class="nocss">Â </span><a shape="rect" href="chap2_mj.html#X7A480B9A795EF264">2.1 <span class="Heading">Various types of text</span></a>
+</span>
+</div>
+<div class="ContSect"><span class="tocline"><span class="nocss">Â </span><a shape="rect" href="chap2_mj.html#X7AA5BF0279938BE0">2.2 <span class="Heading">Formulae</span></a>
+</span>
+</div>
+<div class="ContSect"><span class="tocline"><span class="nocss">Â </span><a shape="rect" href="chap2_mj.html#X833C410D85CF96A4">2.3 <span class="Heading">Crossreferencing</span></a>
+</span>
+</div>
+<div class="ContSect"><span class="tocline"><span class="nocss">Â </span><a shape="rect" href="chap2_mj.html#X7F10E951789D6EDF">2.4 <span class="Heading">Lists and Tables</span></a>
+</span>
+</div>
+<div class="ContSect"><span class="tocline"><span class="nocss">Â </span><a shape="rect" href="chap2_mj.html#X83A355E68485D6D1">2.5 <span class="Heading">Entities and Special Characters</span></a>
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+</div><h3 xmlns="http://www.w3.org/1999/xhtml">2 <span class="Heading">Other Markup</span></h3> </body>
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+ <head>
+ <title>60.5-2 IsGaussianIntegers</title>
+ <meta name="url" content="http://www.gap-system.org/Manuals/doc/ref/TODO#X7BFD33D27BFB7C5A"></meta>
+ </head>
+ <body> <h5 xmlns="http://www.w3.org/1999/xhtml">60.5-2 IsGaussianIntegers</h5><div class="func" xmlns="http://www.w3.org/1999/xhtml"><table width="100%" class="func"><tbody><tr><td colspan="1" rowspan="1" class="tdleft"><code class="func">‣ IsGaussianIntegers</code>( <var class="Arg">obj</var> )</td><td colspan="1" rowspan="1" class="tdright">( category )</td></tr></tbody></table></div><p xmlns="http://www.w3.org/1999/xhtml">is the defining category for the domain <code class="func">GaussianIntegers</code> (<a shape="rect" href="chap60_mj.html#X80BD5EAB879F096E"><span class="RefLink">60.5-1</span></a>).</p><div class="chlinkprevnextbot" xmlns="http://www.w3.org/1999/xhtml"> <a shape="rect" href="chap0_mj.html">[Top of Book]</a> <a shape="rect" href="chap0_mj.html#contents">[Contents]</a>  <a shape="rect" href="chap59_mj.html">[Previous Chapter]</a>  <a shape="rect" href="chap61_mj.html">[Next Chapter]</a> </div><div class="chlinkbot" xmlns="http://www.w3.org/1999/xhtml"><span 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href="chap87_mj.html">87</a> <a shape="rect" href="chapBib_mj.html">Bib</a> <a shape="rect" href="chapInd_mj.html">Ind</a> </div><hr xmlns="http://www.w3.org/1999/xhtml"/><p class="foot" xmlns="http://www.w3.org/1999/xhtml">generated by <a shape="rect" href="http://www.math.rwth-aachen.de/~Frank.Luebeck/GAPDoc">GAPDoc2HTML</a></p> </body>
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+ <title>60.5-1 GaussianIntegers</title>
+ <meta name="url" content="http://www.gap-system.org/Manuals/doc/ref/TODO#X80BD5EAB879F096E"></meta>
+ </head>
+ <body> <h5 xmlns="http://www.w3.org/1999/xhtml">60.5-1 GaussianIntegers</h5><div class="func" xmlns="http://www.w3.org/1999/xhtml"><table width="100%" class="func"><tbody><tr><td colspan="1" rowspan="1" class="tdleft"><code class="func">‣ GaussianIntegers</code></td><td colspan="1" rowspan="1" class="tdright">( global variable )</td></tr></tbody></table></div><p xmlns="http://www.w3.org/1999/xhtml"><code class="func">GaussianIntegers</code> is the ring <math id="-8939236038039800734" display="inline" alttext="?[\sqrt{{-1}}]" class="ltx_Math" xmlns="http://www.w3.org/1998/Math/MathML">
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+</math> of Gaussian integers. This is a subring of the cyclotomic field <code class="func">GaussianRationals</code> (<a shape="rect" href="chap60_mj.html#X82F53C65802FF551"><span class="RefLink">60.1-3</span></a>).</p> </body>
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+ <title>60.5 Gaussians</title>
+ <meta name="url" content="http://www.gap-system.org/Manuals/doc/ref/TODO#X85E9E90D7FE877CC"></meta>
+ </head>
+ <body> <h4 xmlns="http://www.w3.org/1999/xhtml">60.5 <span class="Heading">Gaussians</span></h4> </body>
+ </html>
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