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I have a question about a step used in Szamuely's "Galois Groups and Fundamental Groups" in the excerpt below (see page 120):

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The point of my interest is a statement in Proposition 4.6.1:

Fix a perfect field $k$, $X$ an integral proper normal $k$-curve with function field $K:= K(X)$, and $U ⊂ X$ a nonempty open subset. Denote by $K_s$ a fixed separable closure of $K$.

Define $K_U$ as the composite of all finite subextensions $L|K$ of $K_s$ so that the corresponding finite morphism of proper normal curves is etale above all $P ∈ U$.

My question is why $K_U$ is Galois?

Considerations:

We know that $K_s$ is Galois (by construction/algebra) and there exist a criterion that a subextension field $F \subset K_s$ is Galois if and only if $\sigma(F) \subset F$ for every $\sigma \in Gal(K_s \vert K)$.

By construction of $K_U$ it suffice to show that for every finite subextension $L|K$ of $K_s$ so that the corresponding finite morphism of proper normal curves is etale and an arbitrary $\sigma \in Gal(K_s \vert K)$ the subextension $\sigma(L)$ corresponds also to a finite morphism of proper normal curves which is etale.(*)

But I don't see how to show that.

Remark: the correspondence between finite field extensions of $K$ and integral proper normal curves which I mean is this one:

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Is etaleness conserved by the isomorphism of fields $\sigma:L \to \sigma(L)$? I'm not sure: is the statement () which I wan't to verify is just a simple consequence of functoriality of the correspondence in $4.5.13$ or does ()require a bit more effort?

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