Galois group

Definition. Let $L/K$ be a field extension . The Galois group of $L/K$ is

a subgroup of the field automorphism group of $L$. With composition, $\mathrm{Gal}(L/K)$ is a group .

When $L/K$ is Galois , $\mathrm{Gal}(L/K)$ encodes intermediate fields via the Galois correspondence .

See also. fixed field , automorphism group .

Examples.

1. $\mathrm{Gal}(\mathbb{C}/\mathbb{R})=\{ \mathrm{id}, \text{complex conjugation}\}\cong C_2$.
2. $\mathrm{Gal}(\mathbb{Q}(\sqrt2)/\mathbb{Q})\cong C_2$, sending $\sqrt2\mapsto \pm\sqrt2$.
3. $\mathrm{Gal}(\mathbb{F}_{p^n}/\mathbb{F}_p)$ is cyclic of order $n$, generated by $x\mapsto x^p$ (see finite-field Galois groups are cyclic ).