Let denote the th cyclotomic field, i.e. the extension of the rational numbers obtained by adjoining the th roots of unity to (where is an integer). It is a Galois extension of with Galois group isomorphic to the multiplicative group of integers modulo . The Stickelberger element (of level or of ) is an element in the group ring and the Stickelberger ideal (of level or of ) is an ideal in the group ring . They are defined as follows. Let denote a primitive th root of unity. The isomorphism from to is given by sending to defined by the relation