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

${\displaystyle \sigma _{a}(\zeta _{m})=\zeta _{m}^{a}}$.

The Stickelberger element of level m is defined as

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

The Stickelberger ideal of level m, denoted I(K_m), is the set of integral multiples of θ(K_m) which have integral coefficients, i.e.

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

More generally, if F be any Abelian number field whose Galois group over ${\displaystyle \mathbb {Q} }$ is denoted G_F, then the Stickelberger element ofF and the Stickelberger ideal ofF 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 ${\displaystyle \mathbb {Q} }$). 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_mσ. The Stickelberger element of F is then defined as

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

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

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

In the special case where F = K_m, the Stickelberger ideal I(K_m) is generated by (a − σ_a)θ(K_m) as a varies over ${\displaystyle \mathbb {Z} }$/m${\displaystyle \mathbb {Z} }$. This not true for general F. ^[2]

Examples

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

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

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

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 ${\displaystyle \mathbb {Z} }$[G_F] is.

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

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

and if J is any fractional ideal of F, then

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

is a principal ideal.

See also

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

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. Vol. 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. (ed.). 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. Vol. 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, 1847 (35): 327–367, doi:10.1515/crll.1847.35.327, S2CID   123230326
• Stickelberger, Ludwig (1890), "Ueber eine Verallgemeinerung der Kreistheilung", Mathematische Annalen, 37 (3): 321–367, doi:10.1007/bf01721360, JFM   22.0100.01, MR   1510649, S2CID   121239748
• Washington, Lawrence (1997), Introduction to Cyclotomic Fields, Graduate Texts in Mathematics, vol. 83 (2 ed.), Berlin, New York: Springer-Verlag, ISBN   978-0-387-94762-4, MR   1421575

External links

• PlanetMath page