Cayley's theorem
For showing this, one may proceed as follows. Choose a field and a finite group . Cayley's theorem says that is (up to isomorphism) a subgroup of the symmetric group on the elements of . Choose indeterminates , one for each element of , and adjoin them to to get the field . Contained within is the field of symmetric rational functions in the . The Galois group of is , by a basic result of Emil Artin. acts on by restriction of action of . If the fixed field of this action is , then, by the fundamental theorem of Galois theory, the Galois group of is .