Let K_m denote the mth
cyclotomic field, i.e. the
extension of the
rational numbers obtained by
adjoining the mth
roots of unity to \mathbb{Q} (where m\ge 2 is an integer). It is a
Galois extension of \mathbb{Q} with
Galois group G_m isomorphic to the
multiplicative group of integers modulo (\mathbb{Z}/m\mathbb{Z})^\times. The
Stickelberger element (of level m or of K_m) is an element in the
group ring \mathbb{Q}[G_m] and the
Stickelberger ideal (of level m or of K_m) is an ideal in the group ring \mathbb{Z}[G_m]. They are defined as follows. Let \zeta_m denote a
primitive mth root of unity. The isomorphism from (\mathbb{Z}/m\mathbb{Z})^\times to G_m is given by sending an element a to \sigma_a defined by the relation \sigma_a(\zeta_m) = \zeta_m^a. The Stickelberger element of level m is defined as \theta(K_m)=\frac{1}{m}\underset{(a,m)=1}{\sum_{a=1}^m}a\cdot\sigma_a^{-1}\in\Q[G_m]. The Stickelberger ideal of level m, denoted I(K_m), is the set of integral multiples of \theta(K_m) which have integral coefficients, i.e. I(K_m)=\theta(K_m)\Z[G_m]\cap\Z[G_m]. More generally, if F be any
Abelian number field whose Galois group over \Q is denoted G_F, then the
Stickelberger element of F and the
Stickelberger ideal of F can be defined. By the
Kronecker–Weber theorem there is an integer m such that F is contained in K_m. Fix the least such m (this is the (finite part of the)
conductor of F over \Q). There is a natural
group homomorphism G_m\to G_F given by restriction, i.e. if \sigma_\in G_m, its image in G_F is its restriction to F denoted \operatorname{res}_m\sigma. The Stickelberger element of F is then defined as \theta(F)=\frac{1}{m}\underset{(a,m)=1}{\sum_{a=1}^m}a\cdot\mathrm{res}_m\sigma_a^{-1}\in\Q[G_F]. The Stickelberger ideal of F, denoted I(F), is defined as in the case of K_m, i.e. I(F)=\theta(F)\Z[G_F]\cap\Z[G_F]. In the special case where F=K_m, the Stickelberger ideal I(K_m) is generated by (a-\sigma_a)\theta(K_m) as a varies over \Z/m\Z. This is not true for general F.
Examples If F is a
totally real field of conductor m, then \theta(F)=\frac{\varphi(m)}{2[F:\Q]}\sum_{\sigma\in G_F}\sigma, where \varphi is the
Euler totient function and [F:\Q] is the
degree of F over \Q. ==Statement of the theorem==