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 transitivesubgroup
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 .

The inverse Galois problem asks whether every finite group is isomorphic to a Galois group for some number field .