Galois group

Definition

Let $L/K$ be a field extension. An automorphism of $L/K$ is a ring isomorphism $\sigma \colon L \to L$ with $\sigma(c) = c$ for every $c \in K$. These automorphisms form a group $\operatorname{Aut}(L/K)$ under composition. If $L/K$ is a Galois extension, this group is the Galois group of $L/K$, written $\operatorname{Gal}(L/K)$.

Remarks

Some authors call $\operatorname{Aut}(L/K)$ the Galois group for every extension $L/K$. The group $\operatorname{Gal}(\mathbb{C}/\mathbb{R})$ has two elements: the identity and complex conjugation. The group $\operatorname{Aut}(\mathbb{Q}(\sqrt[3]{2})/\mathbb{Q})$ is trivial, because an automorphism must send $\sqrt[3]{2}$ to a real cube root of $2$. For a finite extension, $|\operatorname{Aut}(L/K)| \le [L : K]$, by the bound on the order.

Explanations

L1 human

The Galois group collects the symmetries of a bigger field that leave the smaller field unchanged. Each symmetry permutes the roots of every polynomial over the smaller field.