Orbits of elements of the field under the action of a subgroup of field automorphisms

Question

Let $G$ be a subgroup of the group of automorphisms of field $E$, $\mid G\mid = n$. Can we find such $\alpha\in E$ that its orbit under $G$ consists of $n$ different elements? If $E$ is a Galois extension of $F$ then we can prove that there is such $\alpha$ that its orbit is the basis of $E$ but I wonder if a less stronger statement is true in general.

Answer 1

It is a theorem in Galois theory that for any finite group of automorphisms $G$ of a field $E$, the fixed field $F = E^G = \{x \in E : g(x) = x \text{ for all } g \in G\}$ is going to fit the situation in your second sentence: $E/F$ is a finite Galois extension with Galois group $G$. So your "less strong" statement is in fact equivalent to your example of Galois extensions. See Lang's "Undergraduate Algebra" or his graduate "Algebra" textbook for a proof that $E/E^G$ is a Galois extension with Galois group $G$.