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Let $E/F$ be a field extension. Let us say $E/F$ is semi-Galois if $F$ is the fixed field of $\rm{Aut}(E/F)$.

Now consider a field tower $E/F/G$. If $E/G$ is semi-Galois, then is $E/F$ necessarily semi-Galois? What would be a counterexample if this is false?

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Let $F$ be an algebraic closure of $\mathbb{Z}/2\mathbb{Z}$. Then,$F(X)/F$ is semi-Galois while $F(X)/F(X^2)$ is not a semi-Galois.

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Characteristic zero counterexample: $\mathbb{C}(z) \supset \mathbb{C}(z^3-z) \supset \mathbb{C}$.

Verification that $\mathbb{C}(z) / \mathbb{C}$ is semi-Galois: The only periodic rational functions are constants, so $f(z)$ is fixed by $z \mapsto z+1$ if and only $f$ is constant.

Verification that $\mathbb{C}(z) / \mathbb{C}(z^3-z)$ is not semi-Galois Write $t = z^3-z$. This field extension is finite, so Galois and semi-Galois are equivalent. This is a cubic extension with minimial polynomial $x^3-x-t=0$. The discriminant of this polynomial, $4 - 27 t^2$, is not a square, so this cubic extension is not Galois.

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