Finite Orbit under Group of Automorphisms of Field implies Separable over Fixed Field
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Theorem
Let $E$ be a field.
Let $G \le \Aut E$ 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:
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 $F$.
$\blacksquare$