Galois Group

A Galois group of an extension field is the group of automorphisms of such that for every . Thus is fixed. The Galois group of is denoted or .

Let be a rational polynomial of polynomial degree with distinct roots, and let be the splitting field of over , i.e., the smallest subfield of containing all the roots of . Then each element of the Galois group permutes the roots of in a unique way. Thus can be identified with a subgroup of the symmetric group , the group of permutations of the roots of . If is irreducible, then is a transitive subgroup of , i.e., given two roots and of , there exists an element of such that .

The roots of are solvable by radicals iff is a solvable group. Since all subgroups of with are solvable, the roots of all polynomials of polynomial degree up to 4 are solvable by radicals. However, polynomials of polynomial degree 5 or greater are generally not solvable by radicals since (and the alternating group ) are not solvable for .

For an irreducible polynomial over of polynomial degree 3, the Galois group is the alternating group if its polynomial discriminant is the square of a rational number, and the symmetric group otherwise (Milne 2022, p. 49).

For a finite Galois extension field generated by a primitive element , each automorphism is determined by its image of . These images are precisely the roots of the extension field minimal polynomial of over . Thus a primitive-element representation realizes the group through the conjugates of one generator.

The inverse Galois problem asks whether every finite group is isomorphic to a Galois group for some finite Galois extension field . The problem remains open in general. Huang et al. (2026) proved that the Mathieu group M₂₃ occurs in this way, completing the realization of all 26 sporadic groups over .

The Galois group of consists of the identity element and complex conjugation. These functions both take a given real to the same real number.