# Composite of certain Finite Subextensions $L|K$ Galois

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

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:

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?