Galois group

Let $K/F$ be a field extension .

Definition (Galois group). The Galois group of $K/F$, denoted $\mathrm{Gal}(K/F)$, is the set of all field automorphisms $\sigma:K\to K$ that fix $F$ pointwise (i.e., $\sigma(a)=a$ for all $a\in F$). With composition as the operation, $\mathrm{Gal}(K/F)$ is a group (a subgroup of the automorphism group of the field $K$).

When $K/F$ is a finite Galois extension , one has $|\mathrm{Gal}(K/F)|=[K:F]$, and the base field $F$ is the fixed field of $\mathrm{Gal}(K/F)$.

Examples.

1. $\mathrm{Gal}(\mathbb{Q}(\sqrt2)/\mathbb{Q})$ has two elements: the identity and the automorphism sending $\sqrt2\mapsto -\sqrt2$. Thus it is cyclic of order $2$.
2. For finite fields, $\mathrm{Gal}(\mathbb{F}_{p^n}/\mathbb{F}_p)$ is cyclic of order $n$, generated by the Frobenius automorphism $x\mapsto x^p$ (compare finite-field Galois groups are cyclic ).
3. For a cyclotomic extension $\mathbb{Q}(\zeta_n)/\mathbb{Q}$, every automorphism is determined by $\zeta_n\mapsto \zeta_n^a$ with $\gcd(a,n)=1$, giving an isomorphism $\mathrm{Gal}(\mathbb{Q}(\zeta_n)/\mathbb{Q}) \cong (\mathbb{Z}/n\mathbb{Z})^\times.$