Lemma 9.22.5. Let $L/M/K$ be a tower of fields. Assume $L/K$ and $M/K$ are 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 profinite topological groups.

Lemma 9.22.5. Let $L/M/K$ be a tower of fields. Assume $L/K$ and $M/K$ are 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 profinite topological groups.

**Proof.**
This is a reformulation of Lemma 9.22.2.
$\square$

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