Lemma 9.22.2. Let $L/M/K$ be a tower of fields. Assume both $L/K$ and $M/K$ are Galois. Then there is a canonical surjective continuous homomorphism $c : \text{Gal}(L/K) \to \text{Gal}(M/K)$.

Proof. By Lemma 9.15.7 given $\tau : L \to L$ in $\text{Gal}(L/K)$ the restriction $\tau |_ M : M \to M$ is an element of $\text{Gal}(M/K)$. This defines the homomorphism $c$. Continuity follows from the universal property of the topology: the action

$\text{Gal}(L/K) \times M \longrightarrow M,\quad (\tau , x) \longmapsto \tau (x) = c(\tau )(x)$

is continuous as $M \subset L$ and the action $\text{Gal}(L/K) \times L \to L$ is continuous. Hence continuity of $c$ by part (1) of Lemma 9.22.1. Lemma 9.15.7 also shows that the map is surjective. $\square$

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