Galois extension

Let $K/F$ be an algebraic field extension .

Definition (Galois extension). The extension $K/F$ is Galois if it is both

• normal , and
• separable .

If $K/F$ is finite, this is equivalent to saying that $K$ is the splitting field of a separable polynomial $f\in F[x]$. In that case, the Galois group $\mathrm{Gal}(K/F)$ has order $[K:F]$ (see degree equals group order ), and the base field $F$ can be recovered as the fixed field of $\mathrm{Gal}(K/F)$.

The equivalence between “normal + separable” and “Galois” is highlighted in separable + normal ⇒ Galois .

Examples.

1. $\mathbb{Q}(\sqrt2)/\mathbb{Q}$ is Galois: it is normal (splitting field of $x^2-2$) and separable (characteristic $0$).
2. The splitting field of $x^3-2$ over $\mathbb{Q}$ is $K=\mathbb{Q}(\sqrt[3]{2},\zeta_3)$. Since $x^3-2$ is separable in characteristic $0$, this splitting field is Galois over $\mathbb{Q}$.
3. For finite fields, $\mathbb{F}_{p^n}/\mathbb{F}_p$ is Galois; its Galois group is cyclic generated by Frobenius. By contrast, $\mathbb{Q}(\sqrt[3]{2})/\mathbb{Q}$ fails to be Galois because it is not normal, and $\mathbb{F}_p(t^{1/p})/\mathbb{F}_p(t)$ fails because it is not separable.