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).^[1]

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 Q (where m ≥ 2 is an integer). It is a Galois extension of Q with Galois group G_m isomorphic to the multiplicative group of integers modulo m (Z/mZ)^×. The Stickelberger element (of level m or of K_m) is an element in the group ring Q[G_m] and the Stickelberger ideal (of level m or of K_m) is an ideal in the group ring Z[G_m]. They are defined as follows. Let ζ_m denote a primitive mth root of unity. The isomorphism from (Z/mZ)^× to G_m is given by sending a to σ_a defined by the relation

σ_a(ζ_m) = ζ a
m .

The Stickelberger element of level m is defined as

$\theta (K_{m})={\frac {1}{m}}{\underset {(a,m)=1}{\sum _{{a=1}}^{m}}}a\cdot \sigma _{a}^{{-1}}\in {\mathbf {Q}}[G_{m}].$

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.

$I(K_{m})=\theta (K_{m}){\mathbf {Z}}[G_{m}]\cap {\mathbf {Z}}[G_{m}].$

More generally, if F be any abelian number field whose Galois group over 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 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_mσ. The Stickelberger element of F is then defined as

$\theta (F)={\frac {1}{m}}{\underset {(a,m)=1}{\sum _{{a=1}}^{m}}}a\cdot {\mathrm {res}}_{m}\sigma _{a}^{{-1}}\in {\mathbf {Q}}[G_{F}].$

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

$I(F)=\theta (F){\mathbf {Z}}[G_{F}]\cap {\mathbf {Z}}[G_{F}].$

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 Z/mZ. This not true for general F.^[2]

Examples

• If F is a totally real field of conductor m, then^[3]

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

where $\phi$ is the Euler totient function and [F : Q] is the degree of F over Q.

Statement of the theorem

Stickelberger's Theorem^[4]

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 Z[G_F] is.

Explicitly, the theorem is saying that if α ∈ Z[G_F] is such that

$\alpha \theta (F)=\sum _{{\sigma \in G_{F}}}a_{\sigma }\sigma \in {\mathbf {Z}}[G_{F}]$

and if J is any fractional ideal of F, then

$\prod _{{\sigma \in G_{F}}}\sigma (J^{{a_{\sigma }}})$

is a principal ideal.

See also

• Gross–Koblitz formula
• Herbrand–Ribet theorem
• Thaine's theorem

Notes

1. Washington 1997, Notes to chapter 6
2. Washington 1997, Lemma 6.9 and the comments following it
3. Washington 1997, §6.2
4. Washington 1997, Theorem 6.10

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. Zbl 1119.11001.
• 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. Zbl 0376.12002.
• 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. MR 1070716.
• Kummer, Ernst (1847), "Über die Zerlegung der aus Wurzeln der Einheit gebildeten complexen Zahlen in ihre Primfactoren", Journal für die Reine und Angewandte Mathematik, 35: 327–367, doi:10.1515/crll.1847.35.327
• Stickelberger, Ludwig (1890), "Ueber eine Verallgemeinerung der Kreistheilung", Mathematische Annalen, 37 (3): 321–367, doi:10.1007/bf01721360, JFM 22.0100.01, MR 1510649
• 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, MR 1421575

External links

• PlanetMath page