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

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Theorem

Let $E$ be a field.

Let $G \leq \operatorname{Aut}(E)$ be a subgroup of its automorphism group.

Let $F = \operatorname{Fix}_E(G)$ 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:

$p(x) = \displaystyle\prod_{\beta \in \Lambda} (x - \beta)$

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 $F$.

$\blacksquare$