The Stacks project

Lemma 10.144.3. Let $R$ be a ring. Let $\mathfrak p$ be a prime of $R$. Let $L/\kappa (\mathfrak p)$ be a finite separable field extension. There exists an étale ring map $R \to R'$ together with a prime $\mathfrak p'$ lying over $\mathfrak p$ such that the field extension $\kappa (\mathfrak p')/\kappa (\mathfrak p)$ is isomorphic to $\kappa (\mathfrak p) \subset L$.

Proof. By the theorem of the primitive element we may write $L = \kappa (\mathfrak p)[\alpha ]$. Let $\overline{f} \in \kappa (\mathfrak p)[x]$ denote the minimal polynomial for $\alpha $ (in particular this is monic). After replacing $\alpha $ by $c\alpha $ for some $c \in R$, $c\not\in \mathfrak p$ we may assume all the coefficients of $\overline{f}$ are in the image of $R \to \kappa (\mathfrak p)$ (verification omitted). Thus we can find a monic polynomial $f \in R[x]$ which maps to $\overline{f}$ in $\kappa (\mathfrak p)[x]$. Since $\kappa (\mathfrak p) \subset L$ is separable, we see that $\gcd (\overline{f}, \overline{f}') = 1$. Hence there is an element $\gamma \in L$ such that $\overline{f}'(\alpha ) \gamma = 1$. Thus we get a $R$-algebra map

\begin{eqnarray*} R[x, 1/f']/(f) & \longrightarrow & L \\ x & \longmapsto & \alpha \\ 1/f' & \longmapsto & \gamma \end{eqnarray*}

The left hand side is a standard étale algebra $R'$ over $R$ and the kernel of the ring map gives the desired prime. $\square$

Comments (0)

Post a comment

Your email address will not be published. Required fields are marked.

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.

In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following input field. As a reminder, this is tag 00UD. Beware of the difference between the letter 'O' and the digit '0'.