Galois group of a polynomial

Definition

Let $K$ be a field and let $f \in K[x]$ be nonzero. Let $L$ be a splitting field of $f$ over $K$. The Galois group of $f$ over $K$ is the group $\operatorname{Aut}(L/K)$ of the automorphisms of $L/K$.

Remarks

By the uniqueness of splitting fields, the group does not depend on the choice of $L$, up to isomorphism. Each automorphism permutes the distinct roots $r_1, \dots, r_m$ of $f$ in $L$, and it is determined by this permutation, because the roots generate $L$. So the group is isomorphic to a subgroup of the symmetric group $S_m$. The Galois group of $x^2 - 2$ over $\mathbb{Q}$ has order $2$. The Galois group of $x^3 - 2$ over $\mathbb{Q}$ is isomorphic to $S_3$. If $f$ is separable, then $L/K$ is a Galois extension.

Explanations

L1 human

The Galois group of a polynomial lists the ways to permute its roots that respect all the algebraic relations between them. It measures how complicated the polynomial is. For example, a polynomial over $\mathbb{Q}$ can be solved by radicals exactly when its Galois group is solvable.