use fancythm. remove \npr
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@ -773,7 +773,7 @@ In general, these inequalities may be strict.
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\subsection{Transcendence degree}
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\subsection{Transcendence degree}
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\subsubsection{Matroids}
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\subsubsection{Matroids}
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\begin{definition}[Hull operator]
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\begin{definition}[Hull operator]
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\npr
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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
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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
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\begin{enumerate}
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\begin{enumerate}
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\item[H1] $\A A \in \cP(X) ~ A \se \cH(A)$.
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\item[H1] $\A A \in \cP(X) ~ A \se \cH(A)$.
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@ -806,7 +806,7 @@ In general, these inequalities may be strict.
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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 $T$.}
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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 $T$.}
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Then $\cH$ is a matroidal hull operator.
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Then $\cH$ is a matroidal hull operator.
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\end{lemma}
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\end{lemma}
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\begin{proof}\npr
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\begin{proof}
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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).
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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).
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Let $x,y \in L$, $T \se L$ and $x \in \cH(T \cup \{y\}) \sm \cH(T)$. We have to show that $y \in \cH(T \cup \{x\}) \sm \cH(T)$.
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Let $x,y \in L$, $T \se L$ and $x \in \cH(T \cup \{y\}) \sm \cH(T)$. We have to show that $y \in \cH(T \cup \{x\}) \sm \cH(T)$.
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@ -3,7 +3,7 @@
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\RequirePackage{mkessler-math}
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\RequirePackage{mkessler-math}
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\RequirePackage[english, numberall]{mkessler-fancythm}
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\RequirePackage[number in = section]{fancythm}
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\RequirePackage{hyperref}
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\RequirePackage{hyperref}
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\RequirePackage[english, index]{mkessler-vocab}
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\RequirePackage[english, index]{mkessler-vocab}
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\RequirePackage{mkessler-hypersetup}
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\RequirePackage{mkessler-hypersetup}
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@ -36,12 +36,12 @@
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\DeclareMathOperator{\hght}{ht}
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\DeclareMathOperator{\hght}{ht}
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\newcommand{\Wlog}{W.l.o.g. }
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\newcommand{\Wlog}{W.l.o.g. }
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\newcommand{\fm}{\ensuremath\mathfrak{m}}
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\newcommand{\Vspec}{\ensuremath V_{\mathbb{S}}}%\Spec}}
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\newcommand{\Vspec}{\ensuremath V_{\mathbb{S}}}%\Spec}}
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\newcommand{\Vs}{\ensuremath V_{\mathbb{S}}}%\Spec}}
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\newcommand{\Vs}{\ensuremath V_{\mathbb{S}}}%\Spec}}
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\newcommand{\Va}{\ensuremath V_{\mathbb{A}}}%\Spec}}
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\newcommand{\Va}{\ensuremath V_{\mathbb{A}}}%\Spec}}
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\newcommand{\Vp}{\ensuremath V_{\mathbb{P}}}%\Spec}}
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\newcommand{\Vp}{\ensuremath V_{\mathbb{P}}}%\Spec}}
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\newcommand{\Pn}{\bP^n}%\Spec}}
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\newcommand{\Pn}{\bP^n}%\Spec}}
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\newcommand{\Span}[1]{\langle#1\rangle}
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\newcommand{\Span}[1]{\langle#1\rangle}
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\newcommand{\npr}{\footnote{Not relevant for the exam.}}
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\newcommand{\npr}{\footnote{Not relevant for the exam.}}
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\newcommand{\limrel}{\footnote{Limited relevance for the exam.}} % may appear in 3x questions
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\newcommand{\limrel}{\footnote{Limited relevance for the exam.}} % may appear in 3x questions
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