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Fundamental theorem of Galois theory

For a finite Galois extension L/K, intermediate fields correspond to subgroups of Gal(L/K).

Fundamental theorem of Galois theory

Let $L/K$ be a finite Galois extension , and write

G = \mathrm{Gal}(L/K)

for its Galois group .

For a subgroup $H\le G$ , the fixed field of $H$ is

L^H=\{x\in L:\sigma(x)=x\text{ for all }\sigma\in H\}.

For an intermediate field $K\subseteq E\subseteq L$ , write

\mathrm{Gal}(L/E)=\{\sigma\in G:\sigma|_E=\mathrm{id}_E\}.

Theorem (Fundamental theorem of Galois theory). The assignments

H \longmapsto L^H \qquad\text{and}\qquad E \longmapsto \mathrm{Gal}(L/E)

give an inclusion-reversing bijection between subgroups $H\le G$ and intermediate fields $E$ with $K\subseteq E\subseteq L$ . Under this correspondence:

$[L:E]=|\,\mathrm{Gal}(L/E)\,|$ and $[E:K]=[G:\mathrm{Gal}(L/E)]$ .
$E/K$ is Galois if and only if $\mathrm{Gal}(L/E)\trianglelefteq G$ is a normal subgroup; in that case there is a natural isomorphism $\mathrm{Gal}(E/K)\cong G/\mathrm{Gal}(L/E).$

This is the conceptual content packaged in the explicit Galois correspondence .

Examples

$L=\mathbb{Q}(\sqrt2)$ over $K=\mathbb{Q}$ .
Then $G\cong C_2$ has exactly two subgroups, so the only intermediate fields are $\mathbb{Q}$ and $\mathbb{Q}(\sqrt2)$ .
$L=\mathbb{Q}(\zeta_8)$ over $\mathbb{Q}$ , where $\zeta_8$ is a primitive 8th root of unity (see cyclotomic extensions ).
Here $G\cong (\mathbb{Z}/8\mathbb{Z})^\times\cong C_2\times C_2$ , which has three distinct index-2 subgroups. Correspondingly, $L/\mathbb{Q}$ has three distinct quadratic intermediate fields.
Finite fields: $L=\mathbb{F}_{p^n}$ over $K=\mathbb{F}_p$ .
The extension is Galois, and $G$ is cyclic of order $n$ generated by Frobenius ; see the cyclic Galois group of finite fields .
Subgroups of a cyclic group correspond to divisors of $n$ , so the intermediate fields are exactly $\mathbb{F}_{p^d}$ for $d\mid n$ .