The Stacks project

\begin{equation*} \DeclareMathOperator\Coim{Coim} \DeclareMathOperator\Coker{Coker} \DeclareMathOperator\Ext{Ext} \DeclareMathOperator\Hom{Hom} \DeclareMathOperator\Im{Im} \DeclareMathOperator\Ker{Ker} \DeclareMathOperator\Mor{Mor} \DeclareMathOperator\Ob{Ob} \DeclareMathOperator\Sh{Sh} \DeclareMathOperator\SheafExt{\mathcal{E}\mathit{xt}} \DeclareMathOperator\SheafHom{\mathcal{H}\mathit{om}} \DeclareMathOperator\Spec{Spec} \newcommand\colim{\mathop{\mathrm{colim}}\nolimits} \newcommand\lim{\mathop{\mathrm{lim}}\nolimits} \newcommand\Qcoh{\mathit{Qcoh}} \newcommand\Sch{\mathit{Sch}} \newcommand\QCohstack{\mathcal{QC}\!\mathit{oh}} \newcommand\Cohstack{\mathcal{C}\!\mathit{oh}} \newcommand\Spacesstack{\mathcal{S}\!\mathit{paces}} \newcommand\Quotfunctor{\mathrm{Quot}} \newcommand\Hilbfunctor{\mathrm{Hilb}} \newcommand\Curvesstack{\mathcal{C}\!\mathit{urves}} \newcommand\Polarizedstack{\mathcal{P}\!\mathit{olarized}} \newcommand\Complexesstack{\mathcal{C}\!\mathit{omplexes}} \newcommand\Pic{\mathop{\mathrm{Pic}}\nolimits} \newcommand\Picardstack{\mathcal{P}\!\mathit{ic}} \newcommand\Picardfunctor{\mathrm{Pic}} \newcommand\Deformationcategory{\mathcal{D}\!\mathit{ef}} \end{equation*}

Lemma 10.153.3. Let $R$ be a ring. Let $\mathfrak p \subset R$ be a prime and let $\kappa (\mathfrak p) \subset L$ be a finite extension of fields. Then there exists a finite free ring map $R \to S$ such that $\mathfrak q = \mathfrak pS$ is prime and $\kappa (\mathfrak p) \subset \kappa (\mathfrak q)$ is isomorphic to the given extension $\kappa (\mathfrak p) \subset L$.

Proof. By induction of the degree of $\kappa (\mathfrak p) \subset L$. If the degree is $1$, then we take $R = S$. In general, if there exists a sub extension $\kappa (\mathfrak p) \subset L' \subset L$ then we win by induction on the degree (by first constructing $R \subset S'$ corresponding to $L'/\kappa (\mathfrak p)$ and then construction $S' \subset S$ corresponding to $L/L'$). Thus we may assume that $L \supset \kappa (\mathfrak p)$ is generated by a single element $\alpha \in L$. Let $X^ d + \sum _{i < d} a_ iX^ i$ be the minimal polynomial of $\alpha $ over $\kappa (\mathfrak p)$, so $a_ i \in \kappa (\mathfrak p)$. We may write $a_ i$ as the image of $f_ i/g$ for some $f_ i, g \in R$ and $g \not\in \mathfrak p$. After replacing $\alpha $ by $g\alpha $ (and correspondingly replacing $a_ i$ by $g^{d - i}a_ i$) we may assume that $a_ i$ is the image of some $f_ i \in R$. Then we simply take $S = R[x]/(x^ d + \sum f_ ix^ i)$. $\square$


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