Separable + Normal ⇔ Galois (Finite Case)

Theorem.
Let $L/K$ be a finite extension. Then $L/K$ is a Galois extension if and only if it is both:

• separable , and
• normal .

When these conditions hold, the Galois group has order $|\mathrm{Gal}(L/K)|=[L:K]$ (see degree equals group order ).

Examples.

1. $\mathbb{Q}(\sqrt2)/\mathbb{Q}$ is separable (char $0$) and normal (splitting field of $x^2-2$), hence Galois.
2. $\mathbb{Q}(\sqrt[3]{2})/\mathbb{Q}$ is separable but not normal, hence not Galois.
3. $\mathbb{F}_{p^n}/\mathbb{F}_p$ is separable and normal (splitting field of $x^{p^n}-x$), hence Galois.

Related. normality via splitting fields , separability via distinct roots .