The Stacks project

Lemma 9.21.8. Let $L/M/K$ be a tower of fields. Assume $L/K$ and $M/K$ are finite Galois. Then we obtain a short exact sequence

\[ 1 \to \text{Gal}(L/M) \to \text{Gal}(L/K) \to \text{Gal}(M/K) \to 1 \]

of finite groups.

Proof. Namely, by Lemma 9.15.7 we see that every element $\tau \in \text{Gal}(L/K)$ induces an element $\tau |_ M \in \text{Gal}(M/K)$ which gives us the homomorphism on the right. The map on the left identifies the left group with the kernel of the right arrow. The sequence is exact because the sizes of the groups work out correctly by multiplicativity of degrees in towers of finite extensions (Lemma 9.7.7). One can also use Lemma 9.15.7 directly to see that the map on the right is surjective. $\square$

Comments (0)

There are also:

  • 4 comment(s) on Section 9.21: Galois theory

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Your email address will not be published.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



View Lemma 9.21.8 as pdf