Lemma 15.14.7. Let $A$ be absolutely integrally closed. Let $\mathfrak p \subset A$ be a prime. Then the local ring $A_\mathfrak p$ is strictly henselian.
Proof. By Lemma 15.14.3 we may assume $A$ is a local ring and $\mathfrak p$ is its maximal ideal. The residue field is algebraically closed by Lemma 15.14.3. Every monic polynomial decomposes completely into linear factors hence Algebra, Definition 10.153.1 applies directly. $\square$
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Your email address will not be published.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).
All contributions are licensed under the GNU Free Documentation License.