Exercise 7.2.5

Proof. In the context of the proof of Theorem 7.2.7, $F\subset K\subset L$, $L∕F$ and $K∕F$ are Galois extensions, and $\sigma ,\tau \in \mathrm{Gal}(L∕F)$.

$\mathit{\sigma K}=K$ by Theorem 7.2.5, thus for all $\alpha \in K$, $\sigma (\alpha )\in K$.

We write here $\sigma {\text{|}}_K:K\to K$ the restriction (and corestriction) of $\sigma$ to $K$, defined by $\sigma {\text{|}}_K(\alpha )=\sigma (\alpha )$.

For all $\alpha \in K$,

Therefore $\mathit{\sigma \tau }{\text{|}}_K=(\sigma \circ \tau ){\text{|}}_K=\sigma {\text{|}}_K\circ \tau {\text{|}}_K=\sigma {\text{|}}_K\tau {\text{|}}_K$: the map

is a group homomorphism. □