# 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 (1847) while the general result is due to Ludwig Stickelberger (1890).

## The Stickelberger element and the Stickelberger ideal

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 ≥ 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 m ($\mathbb {Z}$/m$\mathbb {Z}$)$^×$. 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 ζ$_m$ denote a primitive mth root of unity. The isomorphism from ($\mathbb {Z}$/m$\mathbb {Z}$)$^×$ to G$_m$ is given by sending a to σ$_a$ defined by the relation
    $\sigma _{a}(\zeta _{m})=\zeta _{m}^{a}$.
The Stickelberger element of level m is defined as
    $\theta (K_{m})={\frac {1}{m}}{\underset {(a,m)=1}{\sum _{a=1}^{m}}}a\cdot \sigma _{a}^{-1}\in \mathbb {Q} [G_{m}].$
The Stickelberger ideal of level m, denoted I(K$_m$), is the set of integral multiples of θ(K$_m$) which have integral coefficients, i.e.
    $I(K_{m})=\theta (K_{m})\mathbb {Z} [G_{m}]\cap \mathbb {Z} [G_{m}].$
More generally, if F be any Abelian number field whose Galois group over $\mathbb {Q}$ 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 $\mathbb {Q}$). There is a natural group homomorphism G$_m$ → G$_F$ given by restriction, i.e. if σ ∈ G$_m$, its image in G$_F$ is its restriction to F denoted res$_m$σ. The Stickelberger element of F is then defined as
    $\theta (F)={\frac {1}{m}}{\underset {(a,m)=1}{\sum _{a=1}^{m}}}a\cdot \mathrm {res} _{m}\sigma _{a}^{-1}\in \mathbb {Q} [G_{F}].$
The Stickelberger ideal of F, denoted I(F), is defined as in the case of K$_m$, i.e.
    $I(F)=\theta (F)\mathbb {Z} [G_{F}]\cap \mathbb {Z} [G_{F}].$
In the special case where F = K$_m$, the Stickelberger ideal I(K$_m$) is generated by (a − σ$_a$)θ(K$_m$) as a varies over $\mathbb {Z}$/m$\mathbb {Z}$. This not true for general F.

### Examples

If F is a totally real field of conductor m, then
    $\theta (F)={\frac {\varphi (m)}{2[F:\mathbb {Q} ]}}\sum _{\sigma \in G_{F}}\sigma ,$
where φ is the Euler totient function and [F : $\mathbb {Q}$] is the degree of F over $\mathbb {Q}$.

## Statement of the theorem

   Stickelberger's Theorem Let F be an abelian number field. Then, the Stickelberger ideal of F annihilates the class group of F.
Note that θ(F) itself need not be an annihilator, but any multiple of it in $\mathbb {Z}$[G$_F$] is.
Explicitly, the theorem is saying that if α ∈ $\mathbb {Z}$[G$_F$] is such that
    $\alpha \theta (F)=\sum _{\sigma \in G_{F}}a_{\sigma }\sigma \in \mathbb {Z} [G_{F}]$
and if J is any fractional ideal of F, then
    $\prod _{\sigma \in G_{F}}\sigma \left(J^{a_{\sigma }}\right)$
is a principal ideal.