Stickelberger's theorem


In mathematics, Stickelberger's theorem is a result of algebraic number theory, which gives some information about the Galois module structure of class groups of cyclotomic fields. A special case was first proven by Ernst Kummer while the general result is due to Ludwig Stickelberger.

The Stickelberger element and the Stickelberger ideal

Let denote the th cyclotomic field, i.e. the extension of the rational numbers obtained by adjoining the th roots of unity to . It is a Galois extension of with Galois group isomorphic to the multiplicative group of integers modulo . The Stickelberger element is an element in the group ring and the Stickelberger ideal 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
The Stickelberger element of level is defined as
The Stickelberger ideal of level, denoted, is the set of integral multiples of which have integral coefficients, i.e.
More generally, if be any Abelian number field whose Galois group over is denoted, then the Stickelberger element of and the Stickelberger ideal of can be defined. By the Kronecker–Weber theorem there is an integer such that is contained in. Fix the least such . There is a natural group homomorphism given by restriction, i.e. if, its image in is its restriction to denoted. The Stickelberger element of is then defined as
The Stickelberger ideal of, denoted, is defined as in the case of, i.e.
In the special case where, the Stickelberger ideal is generated by as varies over. This not true for general F.

Examples

If is a totally real field of conductor, then
where is the Euler totient function and is the degree of over ℚ.

Statement of the theorem

Stickelberger's Theorem
Let be an abelian number field. Then, the Stickelberger ideal of annihilates the class group of.

Note that itself need not be an annihilator, but any multiple of it in is.
Explicitly, the theorem is saying that if is such that
and if is any fractional ideal of, then
is a principal ideal.