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 (1847) while the general result is due to Ludwig Stickelberger (1890).
[1] Let Km denote the mth cyclotomic field, i.e. the extension of the rational numbers obtained by adjoining the mth roots of unity to
Let ζm denote a primitive mth root of unity.
)× to Gm is given by sending a to σa defined by the relation The Stickelberger element of level m is defined as The Stickelberger ideal of level m, denoted I(Km), is the set of integral multiples of θ(Km) which have integral coefficients, i.e. More generally, if F be any Abelian number field whose Galois group over
By the Kronecker–Weber theorem there is an integer m such that F is contained in Km.
There is a natural group homomorphism Gm → GF given by restriction, i.e. if σ ∈ Gm, its image in GF is its restriction to F denoted resmσ.
This not true for general F.[2] If F is a totally real field of conductor m, then[3] where φ is the Euler totient function and [F :
Stickelberger's Theorem[4] Let F be an abelian number field.