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 Template:Mvar denote the Template:Mvarth cyclotomic field, i.e. the extension of the rational numbers obtained by adjoining the Template:Mvarth roots of unity to $\mathbb{Q}$ (where Template:Math is an integer). It is a Galois extension of $\mathbb{Q}$ with Galois group Template:Mvar isomorphic to the [[Multiplicative group of integers modulo n|multiplicative group of integers modulo Template:Mvar]] Template:Math. The Stickelberger element (of level Template:Mvar or of Template:Mvar) is an element in the group ring Template:Math and the Stickelberger ideal (of level Template:Mvar or of Template:Mvar) is an ideal in the group ring Template:Math. They are defined as follows. Let Template:Mvar denote a [[primitive root of unity|primitive Template:Mvarth root of unity]]. The isomorphism from Template:Math to Template:Mvar is given by sending Template:Mvar to Template:Mvar defined by the relation

${\sigma }_a({\zeta }_m)={\zeta }_m^a$.

The Stickelberger element of level Template:Mvar 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 Template:Mvar, denoted Template:Math, is the set of integral multiples of Template:Math which have integral coefficients, i.e.

$I(K_m)=\theta (K_m)\mathbb{Z}[G_m]\cap \mathbb{Z}[G_m].$

More generally, if Template:Mvar be any Abelian number field whose Galois group over Template:Math is denoted Template:Mvar, then the Stickelberger element of Template:Mvar and the Stickelberger ideal of Template:Mvar can be defined. By the Kronecker–Weber theorem there is an integer Template:Mvar such that Template:Mvar is contained in Template:Mvar. Fix the least such Template:Mvar (this is the (finite part of the) conductor of Template:Mvar over Template:Math). There is a natural group homomorphism Template:Math given by restriction, i.e. if Template:Math, its image in Template:Mvar is its restriction to Template:Mvar denoted Template:Math. The Stickelberger element of Template:Mvar 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 Template:Mvar, denoted Template:Math, is defined as in the case of Template:Mvar, i.e.

$I(F)=\theta (F)\mathbb{Z}[G_F]\cap \mathbb{Z}[G_F].$

In the special case where Template:Math, the Stickelberger ideal Template:Math is generated by Template:Math as Template:Mvar varies over Template:Math. This not true for general F.

Examples

If Template:Mvar is a totally real field of conductor Template:Mvar, then

$\theta (F)=\frac{\phi (m)}{2[F:\mathbb{Q}]}\sum _{\sigma \in G_F}\sigma ,$

where Template:Mvar is the Euler totient function and Template:Math is the degree of Template:Mvar over $\mathbb{Q}$.

Statement of the theorem

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

Note that Template:Math itself need not be an annihilator, but any multiple of it in Template:Math is.

Explicitly, the theorem is saying that if Template:Math is such that

$\alpha \theta (F)=\sum _{\sigma \in G_F}{a}_{\sigma }\sigma \in \mathbb{Z}[G_F]$

and if Template:Mvar is any fractional ideal of Template:Mvar, 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