Let be an extension field of , denoted , and let be the set of automorphisms of , that is, the set of automorphisms of such that for every , so that is fixed. Then is a group of transformations of , called the Galois group of . The Galois group of is denoted or .
Let be a rational polynomial of degree 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 degree up to 4 are solvable by radicals. However, polynomials of degree 5 or greater are generally not solvable by radicals since (and the alternating group ) are not solvable for .