Commit 09ea1c2a by Xin

### add 7, validated

parent 7f51d469
 \begin{mhmodnl}[creators=Xin]{extension-degree}{zhs} \begin{definition} 令$\fieldExtension\cL\cK$为一个\mtrefi[subfield?field-extension]{域扩张}，则其\defi[name=degree]{次数}$\extensionDegree\cL\cK$定义为将$\cL$的\mtrefi[dimension?dimension]{维度}视作$\cK$上的\mtrefi[vector-space?vector-space]{向量空间}。 \end{definition} \end{mhmodnl} %%% Local Variables: %%% mode: latex %%% TeX-master: t %%% End:
 \begin{mhmodnl}[creators=Xin]{field}{zhs} \begin{definition} \defi[name=field]{域}是其\mtrefi[ring?multiplication]{乘法}是\mtrefi[commutative?commutative]{可交换的}的\mtrefi[ring?ring]{环}，并且除了\mtrefi[ring?zero]{零元}外皆有\defi[name=multiplicative-inverse]{乘法可逆元}$\fieldinv{x}$。称$\fundefeq{a,b}{\fielddiv{a}b}{\rmul{a}{\fieldinv{b}}}$为\defi[name=division]{除法}。 \end{definition} \end{mhmodnl} %%% Local Variables: %%% mode: latex %%% TeX-master: t %%% End:
 \begin{mhmodnl}[creators=Xin]{free-group}{zhs} \begin{definition} 称一个\mtrefi[group?group]{群}$\mvstructure{\magmaset,\magmaopOp,\unitalunit,\groupinvOp}$\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$S$的\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 $\eq{\nsgop{s,\groupinv{t}},\nsgop{s,\groupinv{u},u,\groupinv{t}}}$）。 称$S$为\defi[name=group-generating-set]{生成集合}。 \end{definition} \end{mhmodnl} %%% Local Variables: %%% mode: latex %%% TeX-master: t %%% End: % LocalWords: gimport mhmodnl jusche mtrefi mvstructure magmaset magmaopOp groupinvOp % LocalWords: defi subsupset groupinv groupinv groupinv defii
 \begin{mhmodnl}[creators=Xin]{function-zero}{zhs} \begin{definition} 令$K$为一个\mtrefi[ring?ring]{环}，$\inset{0}K$为其加法单元，且对于集合$D$有$\fun{f}DK$，则称每一个$\inset{z}{\PreImage{f}0}$为一个$f$的\defi[name=zero]{零点}。 \end{definition} \end{mhmodnl} %%% Local Variables: %%% mode: latex %%% TeX-master: t %%% End: \ No newline at end of file
 \begin{mhmodnl}[creators=Xin]{general-linear-group}{zhs} \begin{definition} \mtrefi[field?field]{域}$F$上次数为$n$的\defiii[name=general-linear-group]{一般线性群}是\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}和\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} %%% Local Variables: %%% mode: latex %%% TeX-master: t %%% End: % LocalWords: gimport gle defiii notatiendum GLgroup trefi trefi trefii \ No newline at end of file
 \begin{mhmodnl}[creators=Xin]{generated-subring}{zhs} \begin{definition}[id=generated-subring.def] 令$R$为一个\mtrefi[ring?ring]{环}，$R$的\mtrefi[subring?subring]{子环}的任意\mtrefi[intersection?intersection]{交集}也是$R$的\mtrefi[subring?subring]{子环}。 因此，当$X$是$R$的一个\mtrefi[subsupset?subset]{子集}，所有包含$X$的$R$的\mtrefi[subring?subring]{子环}的\mtrefi[intersection?intersection]{交集}是一个$R$的\mtrefi[subring?subring]{子环}$S$。 $S$是$R$上包含$X$的最小\mtrefi[subring?subring]{子环}。称$S$为$X$的\defi[name=generated-subring]{生成子环}。 当$\eq{X,R}$，则称环$R$由$X$\defi[name=generated]{生成}。 \end{definition} \end{mhmodnl} %%% Local Variables: %%% mode: latex %%% TeX-master: t %%% End: % LocalWords: gimport \ No newline at end of file
 \begin{mhmodnl}[creators=Xin]{generating-set-group}{zhs} \begin{definition}[id=name.def] 称一个\mtrefi[group?group]{群} $\mvstructure{G,\mathord\circ}$由\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$\mathord\circ$）。 若$\sseteq{S}G$，则有$\gensubgroup{S}$，$S$\defi[name=generated-subgroup]{生成子群}是$S$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}。 若$\eq{G,\gensubgroup{S}}$，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} %%% Local Variables: %%% mode: latex %%% TeX-master: t %%% End: % LocalWords: gimport gle defii trefi trefii gensubgroup defiii eq defi \ No newline at end of file
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