Fundamental Theorem of Galois Theory

Theorem (FTGT).
Let $L/K$ be a finite Galois extension with Galois group $G=\mathrm{Gal}(L/K)$. There is an inclusion-reversing bijection

given by

the fixed field of $H$.

It satisfies:

• $[L:E]=|\,\mathrm{Gal}(L/E)\,|$ and $[E:K]=[G:\mathrm{Gal}(L/E)]$ (compare degree = group order ).
• $E/K$ is Galois iff $\mathrm{Gal}(L/E)\trianglelefteq G$, and then $\mathrm{Gal}(E/K)\cong G/\mathrm{Gal}(L/E)$.

Examples.

1. $L=\mathbb{Q}(\sqrt2)$, $G\cong C_2$: subgroups are $\{1\}$ and $G$, corresponding to fields $L$ and $\mathbb{Q}$.
2. $L=\mathbb{F}_{p^n}$ over $K=\mathbb{F}_p$: $G$ is cyclic of order $n$ (see finite-field Galois groups ); subgroups correspond to subfields $\mathbb{F}_{p^d}$ with $d\mid n$.
3. For the splitting field of $x^3-2$ over $\mathbb{Q}$, $G\cong S_3$; intermediate fields correspond to subgroups of $S_3$.

Related. Galois correspondence (core bijection) , Artin's theorem on fixed fields .