Lemma 10.34.12. Let $k$ be a field. Let $S$ be a $k$-algebra. For any field extension $k \subset K$ whose cardinality is larger than the cardinality of $S$ we have
for every maximal ideal $\mathfrak m$ of $S_ K$ the field $\kappa (\mathfrak m)$ is algebraic over $K$, and
$S_ K$ is a Jacobson ring.
Choose $k \subset K$ such that the cardinality of $K$ is greater than the cardinality of $S$. Since the elements of $S$ generate the $K$-algebra $S_ K$ we see that Theorem 10.34.11 applies.
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Your email address will not be published.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).