Galois extension

Definition. An algebraic extension $L/K$ is Galois if it is both

• normal , and
• separable .

In this case the Galois group $\mathrm{Gal}(L/K)$ controls the field via the fundamental theorem of Galois theory . For finite Galois extensions, one has $|\mathrm{Gal}(L/K)|=[L:K]$ (see degree equals group order ).

See also. fixed field , Galois correspondence .

Examples.

1. $\mathbb{C}/\mathbb{R}$ is Galois; $\mathrm{Gal}(\mathbb{C}/\mathbb{R})\cong C_2$.
2. $\mathbb{F}_{p^n}/\mathbb{F}_p$ is Galois (indeed cyclic), generated by the Frobenius automorphism.
3. The splitting field of $x^3-2$ over $\mathbb{Q}$, namely $\mathbb{Q}(\sqrt[3]{2},\zeta_3)$, is Galois over $\mathbb{Q}$.