Theorem. more statements of the Fundamental Theorem [2303]
Theorem. more statements of the Fundamental Theorem [2303]
Let $K / E$ be a finite Galois extension.
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(Inclusion reversing) Suppose that $E_{1}, E_{2}$ are subfields of $K$ containing $F$ and $H_{1}, H_{2}$ are subgroups of $G = \mathrm{Gal}(K / F)$. Then $$\begin{aligned}E_{1} \subseteq E_{2} &\iff \mathrm{Gal}(K / E_{1}) \geq \mathrm{Gal}(K / E_{2}), \\ H_{1} \leq H_{2} &\iff K^{H_{1}} \supseteq K^{H_{2}}\end{aligned}$$
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For all $H \leq G$, $K / K^{H}$ is Galois with $\mathrm{Gal}(K / K^{H}) = H$.
For all $F \subseteq E \subseteq K$, $K / E$ is Galois with $K^{\mathrm{Gal}(K / E)} = E$.
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Given $H \leq G = \mathrm{Gal}(K / F)$, $K^{H} / F$ is Galois if and only if $H \trianglelefteq G$. If this is the case, then $\mathrm{Gal}(K^{H} / F) \cong G / H$.
Given $F \subseteq E \subseteq K$, $E / F$ is Galois if and only if $\mathrm{Gal}(K / E) \trianglelefteq G = \mathrm{Gal}(K / F)$. If this is the case, then $\mathrm{Gal}(E / F) \cong G / \mathrm{Gal}(K / E)$.