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 $m$th cyclotomic field, i.e. the extension of the rational numbers obtained by adjoining the $m$th 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 $m$th 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
 * $$\theta(K_m)=\frac{1}{m}\underset{(a,m)=1}{\sum_{a=1}^m}a\cdot\sigma_a^{-1}\in\Q[G_m].$$

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.
 * $$I(K_m)=\theta(K_m)\Z[G_m]\cap\Z[G_m].$$

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
 * $$\theta(F)=\frac{1}{m}\underset{(a,m)=1}{\sum_{a=1}^m}a\cdot\mathrm{res}_m\sigma_a^{-1}\in\Q[G_F].$$

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 $(a − σ_{a})θ(K_{m})$ 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
 * $$\theta(F)=\frac{\varphi(m)}{2[F:\Q]}\sum_{\sigma\in G_F}\sigma,$$

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
 * $$\alpha\theta(F)=\sum_{\sigma\in G_F}a_\sigma\sigma\in\Z[G_F]$$

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.