Point of confusion in “Topological Representations of Algebras”

Background

I'm reading the article "Topological Representations of Algebras" by Arens, Kaplansky. In the proof of Theorem 6.1 we have the following situation: $X$ is a Stone space, $X_\alpha$ is a family of closed subsets, $K$ is a field with an algebraic closure $M$ and separable closure $L \subseteq M$, $L_\alpha$ are the intermediate fields of $M/L$, $R$ is the ring of continuous functions $X \to K^{alg}$ which map $X_\alpha$ to $K_\alpha$, $G = Aut(L/K)$ is the absolute Galois group of $K$, which may be identified with $Aut(M/K)$. Now, why does $G$ act on $R$ via $(g f)(x) = g f(x)$?

Question

Why is this action well-defined? Don't we need that $L_\alpha / K$ is normal for that? But I doubt that this is true for every algebraic extension of $L$.

There are several other problems in the proof, which arise from the fact that the isomorphism with the ring of study with $R$ is not canonical, but perhaps I'm just missing something ...

Edit: I wonder why nobody answers ... is it too easy? :-)

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Edited the title per an old discussion on meta (tea.mathoverflow.net/discussion/419/is-this-an-error-in-a-paper/…) which seemed to conclude that it's better not to phrase MO questions in ways that imply that a paper is wrong, rather to phrase them in a way that says you have a question about a particular point in a paper. –  Noah Snyder Jun 12 '10 at 15:39