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Question. Let $E$ be a field and let $G=\operatorname{Aut}(E)$ be the group of ring automorphisms. Let $F=\operatorname{Fix}(G)$ be the fixed field of $G$. Is $E$ Galois over $F$? Is $G=\operatorname{Gal}(E/F)$?

Let $p(x)\in F[x]$ be irreducible over $F$ and let $$p(x)=q_1(x)...q_k(x)$$be the factorization of $p(x)$ in $E$ into irreducible factors $q_1(x),...,q_k(x)$. Since any automorphism $\sigma: E\to E$ fixes the coefficient of $p(x)$, $$q_1(x)...q_k(x)=p(x)=(\sigma\cdot p)(x)=(\sigma\cdot q_1)(x)...(\sigma\cdot q_k)(x)$$where $\sigma$ acts on the polynomial by evaluating its coefficient. As $F[x]$ is a UFD, $\sigma$ permutes the irreducible factors $q_1,...,q_k$. Thus once I showed $\operatorname{Aut}(E)$ acts transitively on $q_1,...,q_k$, then $q_1,...,q_k$ must have the same degree.

In particular when $p(x)$ has a root $\alpha\in E$, i.e., when $p(x)$ is the minimal polynomial of some element in $E$, then $p(x)$ must split over $E$. This would prove that $E/F$ is normal.(But I still haven't shown the transitivity)

The equality $G=\operatorname{Gal}(E/F)$ is clear. In general I do not trust $E/F$ to be separable, but I am struggling to come up with a counterexample.

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    $\begingroup$ This is how E. Artin introduces Galois extensions in his lovely book on Galois Theory. Take a field F and a finite group G of automorphisms F. If E is the fixed field, then F/E is Galois with Galois group G. Later he shows that this is equivalent to the finite, normal, separable characterisation $\endgroup$ Commented May 19 at 15:10
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    $\begingroup$ @AndrewHubery That's a classical result in Galois theory. But I guess OP doesn't assume $G$ is finite. $\endgroup$
    – Mark
    Commented May 19 at 15:20
  • $\begingroup$ Let $E=K(T)$, where $T$ is an indeterminate over $K.$ Assume that $K$ is infinite. Then $G=Aut(E)$ contains $H=Aut(E/K)$. But if $K$ is infinite, then it is known that $E^H=K$ (I do not have any reference at the moment). Hence $E/E^H$ is transcendental, and this , so is $E/E^G$. In particular,$E/E^G$ is not Galois. $\endgroup$
    – GreginGre
    Commented May 19 at 17:10
  • $\begingroup$ For infinite groups there is a lemma by Rosenlicht in ‘Automorphisms of function fields’ TAMS 79 (1955), p4. It says that if F is a field, G any group of automorphisms, and E the fixed field, then F/E is separable generated. $\endgroup$ Commented May 19 at 17:52

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