Stickelberger's theorem

Short description: Gives information about the Galois module structure of class groups of cyclotomic fields

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 (Kummer|1847}}|1847) while the general result is due to Ludwig Stickelberger (Stickelberger|1890}}|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 [math]\displaystyle{ \mathbb{Q} }[/math] (where m ≥ 2 is an integer). It is a Galois extension of [math]\displaystyle{ \mathbb{Q} }[/math] with Galois group G_m isomorphic to the multiplicative group of integers modulo m ([math]\displaystyle{ \mathbb{Z} }[/math]/m[math]\displaystyle{ \mathbb{Z} }[/math])^×. The Stickelberger element (of level m or of K_m) is an element in the group ring [math]\displaystyle{ \mathbb{Q} }[/math][G_m] and the Stickelberger ideal (of level m or of K_m) is an ideal in the group ring [math]\displaystyle{ \mathbb{Z} }[/math][G_m]. They are defined as follows. Let ζ_m denote a primitive mth root of unity. The isomorphism from ([math]\displaystyle{ \mathbb{Z} }[/math]/m[math]\displaystyle{ \mathbb{Z} }[/math])^× to G_m is given by sending a to σ_a defined by the relation

[math]\displaystyle{ \sigma_a(\zeta_m) = \zeta_m^a }[/math].

The Stickelberger element of level m is defined as

[math]\displaystyle{ \theta(K_m)=\frac{1}{m}\underset{(a,m)=1}{\sum_{a=1}^m}a\cdot\sigma_a^{-1}\in\Q[G_m]. }[/math]

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.

[math]\displaystyle{ I(K_m)=\theta(K_m)\Z[G_m]\cap\Z[G_m]. }[/math]

More generally, if F be any Abelian number field whose Galois group over [math]\displaystyle{ \mathbb{Q} }[/math] 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 [math]\displaystyle{ \mathbb{Q} }[/math]). 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_mσ. The Stickelberger element of F is then defined as

[math]\displaystyle{ \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]. }[/math]

The Stickelberger ideal of F, denoted I(F), is defined as in the case of K_m, i.e.

[math]\displaystyle{ I(F)=\theta(F)\Z[G_F]\cap\Z[G_F]. }[/math]

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 [math]\displaystyle{ \mathbb{Z} }[/math]/m[math]\displaystyle{ \mathbb{Z} }[/math]. This not true for general F.

Examples

If F is a totally real field of conductor m, then

[math]\displaystyle{ \theta(F)=\frac{\varphi(m)}{2[F:\Q]}\sum_{\sigma\in G_F}\sigma, }[/math]

where φ is the Euler totient function and [F : [math]\displaystyle{ \mathbb{Q} }[/math]] is the degree of F over [math]\displaystyle{ \mathbb{Q} }[/math].

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 [math]\displaystyle{ \mathbb{Z} }[/math][G_F] is.

Explicitly, the theorem is saying that if α ∈ [math]\displaystyle{ \mathbb{Z} }[/math][G_F] is such that

[math]\displaystyle{ \alpha\theta(F)=\sum_{\sigma\in G_F}a_\sigma\sigma\in\Z[G_F] }[/math]

and if J is any fractional ideal of F, then

[math]\displaystyle{ \prod_{\sigma\in G_F}\sigma\left(J^{a_\sigma}\right) }[/math]

is a principal ideal.

See also

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

References

• Cohen, Henri (2007). Number Theory – Volume I: Tools and Diophantine Equations. Graduate Texts in Mathematics. 239. Springer-Verlag. pp. 150–170. ISBN 978-0-387-49922-2. 
• Boas Erez, Darstellungen von Gruppen in der Algebraischen Zahlentheorie: eine Einführung
• Fröhlich, A. (1977). "Stickelberger without Gauss sums". in Fröhlich, A.. Algebraic Number Fields, Proc. Symp. London Math. Soc., Univ. Durham 1975. Academic Press. pp. 589–607. ISBN 0-12-268960-7. 
• Ireland, Kenneth; Rosen, Michael (1990). A Classical Introduction to Modern Number Theory. Graduate Texts in Mathematics. 84 (2nd ed.). New York: Springer-Verlag. doi:10.1007/978-1-4757-2103-4. ISBN 978-1-4419-3094-1. 
• Kummer, Ernst (1847), "Über die Zerlegung der aus Wurzeln der Einheit gebildeten complexen Zahlen in ihre Primfactoren", Journal für die Reine und Angewandte Mathematik 1847 (35): 327–367, doi:10.1515/crll.1847.35.327, https://zenodo.org/record/1448852 
• Stickelberger, Ludwig (1890), "Ueber eine Verallgemeinerung der Kreistheilung", Mathematische Annalen 37 (3): 321–367, doi:10.1007/bf01721360, http://gdz.sub.uni-goettingen.de/no_cache/dms/load/img/?IDDOC=27547 
• Washington, Lawrence (1997), Introduction to Cyclotomic Fields, Graduate Texts in Mathematics, 83 (2 ed.), Berlin, New York: Springer-Verlag, ISBN 978-0-387-94762-4 

External links

• PlanetMath page