diff --git a/2021_Algebra_I.tex b/2021_Algebra_I.tex index cbfc43c..d6750cc 100644 --- a/2021_Algebra_I.tex +++ b/2021_Algebra_I.tex @@ -773,7 +773,7 @@ In general, these inequalities may be strict. \subsection{Transcendence degree} \subsubsection{Matroids} \begin{definition}[Hull operator] - \npr + Let $X$ be a set, $\cP(X)$ the power set of $X$. A \vocab{Hull operator} on $X$ is a map $\cP(X) \xrightarrow{\cH} \cP(X)$ such that \begin{enumerate} \item[H1] $\A A \in \cP(X) ~ A \se \cH(A)$. @@ -806,7 +806,7 @@ In general, these inequalities may be strict. Let $L / K$ be a field extension and let $\cH(T)$ be the algebraic closure in $L$ of the subfield of $L$ generated by $K$ and $T$.\footnote{This is the intersection of all subfields of $L$ containing $K \cup T$, or the field of quotients of the sub-$K$-algebra of $L$ generated by $T$.} Then $\cH$ is a matroidal hull operator. \end{lemma} -\begin{proof}\npr +\begin{proof} H1, H2 and F are trivial. For an algebraically closed subfield $K \se M \se L$ we have $\cH(M) = M$. Thus $\cH(\cH(T)) = \cH(T)$ (H3). Let $x,y \in L$, $T \se L$ and $x \in \cH(T \cup \{y\}) \sm \cH(T)$. We have to show that $y \in \cH(T \cup \{x\}) \sm \cH(T)$. diff --git a/algebra.sty b/algebra.sty index 2f8091a..0eb48b1 100644 --- a/algebra.sty +++ b/algebra.sty @@ -3,7 +3,7 @@ \RequirePackage{mkessler-math} -\RequirePackage[english, numberall]{mkessler-fancythm} +\RequirePackage[number in = section]{fancythm} \RequirePackage{hyperref} \RequirePackage[english, index]{mkessler-vocab} \RequirePackage{mkessler-hypersetup} @@ -36,12 +36,12 @@ \DeclareMathOperator{\hght}{ht} \newcommand{\Wlog}{W.l.o.g. } -\newcommand{\fm}{\ensuremath\mathfrak{m}} \newcommand{\Vspec}{\ensuremath V_{\mathbb{S}}}%\Spec}} \newcommand{\Vs}{\ensuremath V_{\mathbb{S}}}%\Spec}} \newcommand{\Va}{\ensuremath V_{\mathbb{A}}}%\Spec}} \newcommand{\Vp}{\ensuremath V_{\mathbb{P}}}%\Spec}} \newcommand{\Pn}{\bP^n}%\Spec}} + \newcommand{\Span}[1]{\langle#1\rangle} \newcommand{\npr}{\footnote{Not relevant for the exam.}} \newcommand{\limrel}{\footnote{Limited relevance for the exam.}} % may appear in 3x questions