Commit 4a938f88 by 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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