Galois group as permutation group

A Galois group is abstractly a group of field automorphisms, but it has a vivid concrete face: every automorphism just shuffles the roots of the defining polynomial among themselves. So the Galois group acts as a group of permutations of those finitely many roots — Galois's original picture, and still the most computational way to think about it.

Precisely, let f be a separable polynomial over K with roots r_1, ..., r_n in a splitting field L. Each sigma in Gal(L/K) permutes the roots (since it fixes K it sends roots of f to roots of f), and because the roots generate L this permutation determines sigma completely. This gives a faithful, injective homomorphism Gal(L/K) -> S_n, realizing the Galois group as a subgroup of the symmetric group on n letters. The image is determined only up to relabeling the roots, that is, up to conjugacy in S_n.

Two structural facts make this picture powerful. First, the action is transitive on the roots exactly when f is irreducible over K — so irreducibility translates into transitivity. Second, the discriminant, resolvents, and cycle-type information (from factoring f modulo primes, via Dedekind's theorem) let you locate the Galois group precisely among the transitive subgroups of S_n. This is how one computes that, say, x^5 - x - 1 has Galois group S_5 and is therefore not solvable by radicals.