The Stacks project

Lemma 10.34.2. Let $R$ be a ring. Let $K$ be a field. If $R \subset K$ and $K$ is of finite type over $R$, then there exists an $f \in R$ such that $R_ f$ is a field, and $K/R_ f$ is a finite field extension.

Proof. By Lemma 10.30.2 there exist a nonempty open $U \subset \mathop{\mathrm{Spec}}(R)$ contained in the image $\{ (0)\} $ of $\mathop{\mathrm{Spec}}(K) \to \mathop{\mathrm{Spec}}(R)$. Choose $f \in R$, $f \not= 0$ such that $D(f) \subset U$, i.e., $D(f) = \{ (0)\} $. Then $R_ f$ is a domain whose spectrum has exactly one point and $R_ f$ is a field. Then $K$ is a finitely generated algebra over the field $R_ f$ and hence a finite field extension of $R_ f$ by the Hilbert Nullstellensatz (Theorem 10.34.1). $\square$

Comments (2)

Comment #8306 by William Sun on

Other than overkilling with Chevalley's theorem, here is an more elementary argument. is of finite type over , thus finite and thus integral. Choose generators of over and consider their minimal polynomials with coefficients in . By inverting the product of the (finite number of) denominators, we conclude the generators of are integral elements over . Thus is integral over , and is a field, as desired.

Comment #8930 by on

Dear William, I note that the first step of your proof uses the Hilbert Nullstellensatz (unless I am overlooking something). Now our proof of the Nullstellensatz already uses Chevalley. So this wouldn't avoid using Chevalley. Additionally, there is a step in your proof which uses a lemma that comes after this one, so we can't use that here.

There are also:

  • 4 comment(s) on Section 10.34: Hilbert Nullstellensatz

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Your email address will not be published.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 00FY. Beware of the difference between the letter 'O' and the digit '0'.