Degree of a Finite Galois Extension Equals Group Order

Theorem.
If $L/K$ is a finite Galois extension , then

Equivalently, the Galois group has size equal to the degree .

This is a key numerical input in the fundamental theorem of Galois theory .

Examples.

1. $\mathbb{Q}(\sqrt2)/\mathbb{Q}$ is Galois with group $\{1,\sqrt2\mapsto-\sqrt2\}$, so $[L:K]=2=|G|$.
2. $\mathbb{F}_{p^n}/\mathbb{F}_p$ is Galois and $|\mathrm{Gal}(\mathbb{F}_{p^n}/\mathbb{F}_p)|=n$ (see cyclicity of the finite-field Galois group ).
3. If $L$ is the splitting field over $\mathbb{Q}$ of an irreducible separable cubic with Galois group $S_3$, then $[L:\mathbb{Q}]=6$.

Related. Galois ⇔ separable + normal .