Commit 09ea1c2a authored by Xin's avatar Xin

add 7, validated

parent 7f51d469
\begin{mhmodnl}[creators=Xin]{extension-degree}{zhs}
\begin{definition}
$\fieldExtension\cL\cK$为一个\mtrefi[subfield?field-extension]{域扩张},则其\defi[name=degree]{次数}$\extensionDegree\cL\cK$定义为将$\cL$\mtrefi[dimension?dimension]{维度}视作$\cK$上的\mtrefi[vector-space?vector-space]{向量空间}
\end{definition}
\end{mhmodnl}
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\begin{mhmodnl}[creators=Xin]{field}{zhs}
\begin{definition}
\defi[name=field]{}是其\mtrefi[ring?multiplication]{乘法}\mtrefi[commutative?commutative]{可交换的}\mtrefi[ring?ring]{},并且除了\mtrefi[ring?zero]{零元}外皆有\defi[name=multiplicative-inverse]{乘法可逆元}$\fieldinv{x}$。称$\fundefeq{a,b}{\fielddiv{a}b}{\rmul{a}{\fieldinv{b}}}$\defi[name=division]{除法}
\end{definition}
\end{mhmodnl}
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\begin{mhmodnl}[creators=Xin]{free-group}{zhs}
\begin{definition}
称一个\mtrefi[group?group]{}$\mvstructure{\magmaset,\magmaopOp,\unitalunit,\groupinvOp}$\defi[name=free-group]{自由的},当有$\magmaset$的一个\mtrefi[subsupset?subset]{子集}$S$使得每个$\magmaset$的元素都可被唯一地写成$S$\mtrefi[finite-cardinality?finite]{有穷}多元素和其\mtrefi[group?inverse]{逆元}的乘积(除去易得情况如 $\eq{\nsgop{s,\groupinv{t}},\nsgop{s,\groupinv{u},u,\groupinv{t}}}$)。
$S$\defi[name=group-generating-set]{生成集合}
\end{definition}
\end{mhmodnl}
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\begin{mhmodnl}[creators=Xin]{function-zero}{zhs}
\begin{definition}
$K$为一个\mtrefi[ring?ring]{}$\inset{0}K$为其加法单元,且对于集合$D$$\fun{f}DK$,则称每一个$\inset{z}{\PreImage{f}0}$为一个$f$\defi[name=zero]{零点}
\end{definition}
\end{mhmodnl}
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\begin{mhmodnl}[creators=Xin]{general-linear-group}{zhs}
\begin{definition}
\mtrefi[field?field]{}$F$上次数为$n$\defiii[name=general-linear-group]{一般线性群}\mtrefi[matrix?invertible]{可逆的}$n\times n$\mtrefi[matrix?matrix]{矩阵}的集合,带有普通\mtrefi[matrix?matrix-multiplication]{矩阵乘法}\mtrefi[matrix?matrix-inversion]{矩阵逆}运算。
\end{definition}
\begin{omtext}[type=elaboration,for=generallineargroup.def]
\guse[smglom/arithmetics]{complexnumbers} If $V$ is a
\trefii[vector-space]{vector}{space} over the \trefi[field]{field} $F$, then
$\GLgroup{n}{V}$ is the group of all \term{automorphisms} of $V$.\ednote{this
needs to become a view}
\end{omtext}
\begin{omtext}[type=elaboration,for=generallineargroup.def]
\guse[smglom/manifolds]{liegroup}
\guse[smglom/arithmetics]{complexnumbers}
\guse[smglom/arithmetics]{arithmetics}
If $F = \RealNumbers$ or $F = \ComplexNumbers$,
then $\GLgroup{n}{F}$ is a \trefii[liegroup]{Lie}{group} of dimension
$\power{n}2$. \ednote{this needs to become a view}
\end{omtext}
\end{mhmodnl}
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\begin{mhmodnl}[creators=Xin]{generated-subring}{zhs}
\begin{definition}[id=generated-subring.def]
$R$为一个\mtrefi[ring?ring]{}$R$\mtrefi[subring?subring]{子环}的任意\mtrefi[intersection?intersection]{交集}也是$R$\mtrefi[subring?subring]{子环}
因此,当$X$$R$的一个\mtrefi[subsupset?subset]{子集},所有包含$X$$R$\mtrefi[subring?subring]{子环}\mtrefi[intersection?intersection]{交集}是一个$R$\mtrefi[subring?subring]{子环}$S$
$S$$R$上包含$X$的最小\mtrefi[subring?subring]{子环}。称$S$$X$\defi[name=generated-subring]{生成子环}
$\eq{X,R}$,则称环$R$$X$\defi[name=generated]{生成}
\end{definition}
\end{mhmodnl}
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\begin{mhmodnl}[creators=Xin]{generating-set-group}{zhs}
\begin{definition}[id=name.def]
称一个\mtrefi[group?group]{} $\mvstructure{G,\mathord\circ}$\mtrefi[subsupset?subset]{子集} $\sseteq{S}G$\defi[name=generated]{生成},当且仅当$G$中所有的元素都可以被表示为$S$中的有穷多元素和其的\mtrefi[group?inverse]{逆元}的组合(满足$\mathord\circ$)。
$\sseteq{S}G$,则有$\gensubgroup{S}$$S$\defi[name=generated-subgroup]{生成子群}$S$中所有可被表示为$S$中元素和其\mtrefi[group?inverse]{逆元}的有穷组合的元素的\mtrefi[subgroup?subgroup]{子群}
$\eq{G,\gensubgroup{S}}$,则称$G$是由$S$\defi[name=generates]{生成的}并称$S$中的元素为\defi[name=generator]{生成元}
\end{definition}
\end{mhmodnl}
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