The **Galois group** is a mathematical structure (group) associated to an extension of fields. Given two fields E and F where F (the "base field") is contained in E (the "extension field"), we define Gal(E/F) to be the set of automorphisms of E which fix F. The Galois group of a polynomial f over a field F is the Galois group associated to F(α) where α is a root of f. The group obtained does not depend on the choice of root.

Galois groups are named after Évariste Galois, a French mathematician.