The Stacks project

Lemma 9.14.9 (Multiplicativity). Suppose given a tower of algebraic field extensions $K/E/F$. Then

\[ [K : F]_ s = [K : E]_ s [E : F]_ s \quad \text{and}\quad [K : F]_ i = [K : E]_ i [E : F]_ i \]

Proof. We first prove this in case $K$ is finite over $F$. Since we have multiplicativity for the usual degree (by Lemma 9.7.7) it suffices to prove one of the two formulas. By Lemma 9.14.8 we have $[K : F]_ s = |\mathop{\mathrm{Mor}}\nolimits _ F(K, \overline{F})|$. By the same lemma, given any $\sigma \in \mathop{\mathrm{Mor}}\nolimits _ F(E, \overline{F})$ the number of extensions of $\sigma $ to a map $\tau : K \to \overline{F}$ is $[K : E]_ s$. Namely, via $E \cong \sigma (E) \subset \overline{F}$ we can view $\overline{F}$ as an algebraic closure of $E$. Combined with the fact that there are $[E : F]_ s = |\mathop{\mathrm{Mor}}\nolimits _ F(E, \overline{F})|$ choices for $\sigma $ we obtain the result.

We omit the proof if the extensions are infinite. $\square$

Comments (2)

Comment #6526 by prime235711 on As you see here, a field can have uncountable extension. But the equation makes the separable degree of K over F at most countable

Comment #6581 by on

OK, thanks for pointing this out. I think the first version of this chapter was written in a way where the degree of an extension by definition is either an integer or . But as you ponit out it can be any cardinal and then you can still state this lemma. But for now I have removed the proof in the infinite case as we don't use it anyway. Anybody should feel free to latex up a detailed proof in the infinite case. See changes in this commit.

There are also:

  • 3 comment(s) on Section 9.14: Purely inseparable extensions

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Your email address will not be published.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 09HK. Beware of the difference between the letter 'O' and the digit '0'.