Stickelberger's theorem

Template:Short description 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 ([[#Template:Harvid|1847]]) while the general result is due to Ludwig Stickelberger ([[#Template:Harvid|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 $m\ge 2$ is an integer). It is a Galois extension of $\mathbb{Q}$ with Galois group $G_m$ isomorphic to the [[Multiplicative group of integers modulo n|multiplicative group of integers modulo Template:Mvar]] $(\mathbb{Z}/m\mathbb{Z}{)}^{\times }$. 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 $m$th 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 $$\theta (K_m)=\frac{1}{m}\underset{(a,m)=1}{\sum _{a=1}^m}a\cdot {\sigma }_a^{-1}\in \mathbb{Q}[G_m].$$ 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. $$I(K_m)=\theta (K_m)\mathbb{Z}[G_m]\cap \mathbb{Z}[G_m].$$

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 $G_m\to G_F$ given by restriction, i.e. if ${\sigma }_{\in }{G}_m$, its image in $G_F$ is its restriction to $F$ denoted ${\mathrm{res}}_m\sigma$. The Stickelberger element of $F$ is then defined as $$\theta (F)=\frac{1}{m}\underset{(a,m)=1}{\sum _{a=1}^m}a\cdot {\mathrm{r}\mathrm{e}\mathrm{s}}_m{\sigma }_a^{-1}\in \mathbb{Q}[G_F].$$ 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 $(a-{\sigma }_a)\theta (K_m)$ as $a$ varies over $\mathbb{Z}/m\mathbb{Z}$. This is not true for general $F$.

Examples

If $F$ is a totally real field of conductor $m$, then $$\theta (F)=\frac{\phi (m)}{2[F:\mathbb{Q}]}\sum _{\sigma \in G_F}\sigma ,$$ where $\phi$ 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 $\theta (F)$ itself need not be an annihilator, but any multiple of it in $\mathbb{Z}[G_F]$ is.

Explicitly, the theorem is saying that if $\alpha \in \mathbb{Z}[G_F]$ 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.

See also

• Gross–Koblitz formula
• Herbrand–Ribet theorem
• Thaine's theorem
• Jacobi sum
• Gauss sum

References

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• Boas Erez, Darstellungen von Gruppen in der Algebraischen Zahlentheorie: eine Einführung
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External links

• PlanetMath page