The Galois group Gal(K:F) of a field extension F⊆K is the group Aut_F(K) of field automorphisms of K, which pointwise fix the base field F; the group operation is composition. The Galois group Gal(ℂ:ℝ) has order 2 and is generated by complex conjugation.

If f is a polynomial over F, then the Galois group of f is the Galois group of its splitting field. The Galois group of f acts (see action) on the set of roots of f. A polynomial f over ℚ is solvable by radicals if and only if its Galois group is solvable. The Galois group of x⁵−x−1 is the symmetric group S₅, which is not solvable. See Galois correspondence.