Finite Orbit under Group of Automorphisms of Field implies Separable over Fixed Field

Theorem
Let $E$ be a field.

Let $G \le \Aut E$ be a subgroup of its automorphism group.

Let $F = \map {\operatorname {Fix}_E} G$ be its fixed field.

Let $\alpha \in E$ have a finite orbit under $G$.

Then $\alpha$ is separable over $F$.

Proof
Let $\Lambda$ be the orbit of $\alpha$ under $G$.

By:
 * Minimal Polynomial of Element with Finite Orbit under Group of Automorphisms over Fixed Field in terms of Orbit

the product:
 * $\map p x = \ds \prod_{\beta \in \Lambda} \paren {x - \beta}$

is the minimal polynomial of $\alpha$ over $F$.

By Product of Distinct Monic Linear Polynomials is Separable, $p$ is separable.

Thus $\alpha$ is separable over $F$.