Commit 4a938f88 authored by Michael Kohlhase's avatar Michael Kohlhase

tweaks

parent a06d4629
......@@ -2,7 +2,7 @@
\begin{definition}[id=algebraicclosure.def]
An \defii{algebraic}{closure} of a \trefi[field]{field} is an
\trefii[algebraicextension]{algebraic}{extension} of $K$ which is
\trefii[algebraicallyclosedfield]{algebraically}{closed}
\trefii[algebraicallyclosedfield]{algebraically}{closed}.
\end{definition}
\end{mhmodnl}
%%% Local Variables:
......
\begin{mhmodnl}[creators=cdemirkiran,contributors=miko]{algebraicextension}{en}
\begin{definition}[id=algebraicextension.def]
A \trefii[fieldextension]{field}{extension} $L/K$ is \defi{algebraic} if every
A \trefii[subfield]{field}{extension} $L/K$ is \defi{algebraic} if every
\trefi[element]{element} of $L$ is \trefi[algebraic]{algebraic} over $K$.
\end{definition}
\end{mhmodnl}
......
\begin{modsig}[creators=cdemirkiran,contributors=miko]{algebraicextension}
\gimport[smglom/algebra]{field}
\gimport{fieldextension}
\gimport[smglom/algebra]{subfield}
\gimport[smglom/algebra]{polynomial}
\symi{algebraic}
\end{modsig}
......
......@@ -3,7 +3,7 @@
An \defii{algebraic}{group} is a \trefi[group]{group} and an
\trefiis[algebraicvariety]{algebraic}{variety} such that the \trefii[magma]{operation}
are given by \trefii[regularfunction]{regular}{functions} on the
\trefi[algebraicvariety]{variety}
\trefi[algebraicvariety]{variety}.
\end{definition}
\end{mhmodnl}
%%% Local Variables:
......
\begin{mhmodnl}[creators=cdemirkiran,contributors=miko]{galoisgroup}{en}
\begin{definition}[id=galoisgroup.def]
Assume that $E$ is an \trefi[fieldextension]{extension} of a \trefi[field]{field} $F$.
An \term{automorphism} of $E/F$ is defined to be an \term{automorphism} of $E$ that
fixes $F$ pointwise.
Assume that $E$ is an \atrefii[subfield]{extension}{field}{extension} of a
\trefi[field]{field} $F$. An \term{automorphism} of $E/F$ is defined to be an
\term{automorphism} of $E$ that fixes $F$ pointwise.
The \trefi{set} of all \term{automorphisms} of $E/F$ together with the
\trefi[magma]{operation} of function \trefi[relation-composition]{composition}
forms a \trefi[group]{group} called the \defii{Galois}{group}.
\trefi[magma]{operation} of function \trefi[relation-composition]{composition} forms a
\trefi[group]{group} called the \defii{Galois}{group}.
\end{definition}
\end{mhmodnl}
......
\begin{modsig}[creators=cdemirkiran,contributors=miko]{galoisgroup}
\gimport[smglom/algebra]{field}
\gimport{fieldextension}
\gimport[smglom/algebra]{subfield}
\gimport[smglom/sets]{relation-composition}
\symdef{galoisgroup}[2]{Gal(#1/#2)}
......
\begin{mhmodnl}[creators=cdemirkiran,contributors=miko]{minimalpolynomial}{en}
\begin{definition}[id=minimalpolynomial.def]
Let $E/F$ be a \trefii[fieldextension]{field}{extension}, $\alpha \in E$, the
Let $E/F$ be a \trefii[subfield]{field}{extension}, $\inset\alpha{E}$, the
\defii{minimal}{polynomial} of $alpha$ is the
\trefii[monicpolynomial]{monic}{polynomial} of least \trefi[polynomial]{degree} among
all \trefi[polynomial]{polynomials} in $F[x]$ having $\alpha$ as a \trefi[root]{root}.
......
\begin{modsig}[creators=cdemirkiran,contributors=miko]{minimalpolynomial}
\gimport[smglom/algebra]{field}
\gimport{fieldextension}
\gimport[smglom/algebra]{subfield}
\gimport[smglom/algebra]{monicpolynomial}
\gimport[smglom/algebra]{root}
\symii{minimal}{polynomial}
......
\begin{mhmodnl}[creators=cdemirkiran,contributors=miko]{normalextension}{en}
\begin{definition}[id=normalextension.def]
A \trefii[fieldextension]{field}{extension} $K/F$ is \defi{normal} if $K$ is the
A \trefii[subfield]{field}{extension} $K/F$ is \defi{normal} if $K$ is the
\trefii[splittingfield]{splitting}{field} of every
\trefii[irreduciblepolynomial]{irreducible}{polynomial} $f \in F[x]$ which has at least
one \trefi[root]{root} in $K$.
......
\begin{modsig}[creators=cdemirkiran,contributors=miko]{normalextension}
\gimport{fieldextension}
\gimport[smglom/algebra]{subfield}
\gimport{splittingfield}
\gimport[smglom/algebra]{irreduciblepolynomial}
\gimport[smglom/algebra]{root}
......
\begin{mhmodnl}[creators=cdemirkiran,contributors=miko]{splittingfield}{en}
\begin{definition}[id=splittingfield.def]
A \defii{splitting}{field} of a \trefi[poynomial]{polynomial} $p(x)$ over a
\trefi[field]{field} $K$ is a \trefii[fieldextension]{field}{extension} $L$ of $K$ over
\trefi[field]{field} $K$ is a \trefii[subfield]{field}{extension} $L$ of $K$ over
which $p(x)$ factors into \term{linear factors}
$p(x) = \Prodfromto{i}1{deg(p)}{(x - a_i)}$ such that $a_i$ generate $L$ over $K$.
\end{definition}
......
\begin{modsig}[creators=cdemirkiran,contributors=miko]{splittingfield}
\gimport[smglom/algebra]{field}
\gimport[smglom/algebra]{polynomial}
\gimport{fieldextension}
\gimport[smglom/algebra]{subfield}
\gimport[smglom/numberfields]{product}
\symii{splitting}{field}
\end{modsig}
......
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