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 ${\displaystyle K_{m}}$  denote the ${\displaystyle m}$ th cyclotomic field, i.e. the extension of the rational numbers obtained by adjoining the ${\displaystyle m}$ th roots of unity to ${\displaystyle \mathbb {Q} }$  (where ${\displaystyle m\geq 2}$  is an integer). It is a Galois extension of ${\displaystyle \mathbb {Q} }$  with Galois group ${\displaystyle G_{m}}$  isomorphic to the multiplicative group of integers modulo m ${\displaystyle (\mathbb {Z} /m\mathbb {Z} )^{\times }}$ . The Stickelberger element (of level ${\displaystyle m}$  or of ${\displaystyle K_{m}}$ ) is an element in the group ring ${\displaystyle \mathbb {Q} [G_{m}]}$  and the Stickelberger ideal (of level ${\displaystyle m}$  or of ${\displaystyle K_{m}}$ ) is an ideal in the group ring ${\displaystyle \mathbb {Z} [G_{m}]}$ . They are defined as follows. Let ${\displaystyle \zeta _{m}}$  denote a primitive ${\displaystyle m}$ th root of unity. The isomorphism from ${\displaystyle (\mathbb {Z} /m\mathbb {Z} )^{\times }}$  to ${\displaystyle G_{m}}$  is given by sending an element ${\displaystyle a}$  to ${\displaystyle \sigma _{a}}$  defined by the relation $${\displaystyle \sigma _{a}(\zeta _{m})=\zeta _{m}^{a}.}$$  The Stickelberger element of level ${\displaystyle m}$  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 ${\displaystyle m}$ , denoted ${\displaystyle I(K_{m})}$ , is the set of integral multiples of ${\displaystyle \theta (K_{m})}$  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 ${\displaystyle F}$  be any Abelian number field whose Galois group over ${\displaystyle \mathbb {Q} }$  is denoted ${\displaystyle G_{F}}$ , then the Stickelberger element of ${\displaystyle F}$  and the Stickelberger ideal of ${\displaystyle F}$  can be defined. By the Kronecker–Weber theorem there is an integer ${\displaystyle m}$  such that ${\displaystyle F}$  is contained in ${\displaystyle K_{m}}$ . Fix the least such ${\displaystyle m}$  (this is the (finite part of the) conductor of ${\displaystyle F}$  over ${\displaystyle \mathbb {Q} }$ ). There is a natural group homomorphism ${\displaystyle G_{m}\to G_{F}}$  given by restriction, i.e. if ${\displaystyle \sigma _{\in }G_{m}}$ , its image in ${\displaystyle G_{F}}$  is its restriction to ${\displaystyle F}$  denoted ${\displaystyle \operatorname {res} _{m}\sigma }$ . The Stickelberger element of ${\displaystyle F}$  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 ${\displaystyle F}$ , denoted ${\displaystyle I(F)}$ , is defined as in the case of ${\displaystyle K_{m}}$ , i.e. $${\displaystyle I(F)=\theta (F)\mathbb {Z} [G_{F}]\cap \mathbb {Z} [G_{F}].}$$ 

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

Examples

If ${\displaystyle F}$  is a totally real field of conductor ${\displaystyle m}$ , then^[3] $${\displaystyle \theta (F)={\frac {\varphi (m)}{2[F:\mathbb {Q} ]}}\sum _{\sigma \in G_{F}}\sigma ,}$$  where ${\displaystyle \varphi }$  is the Euler totient function and ${\displaystyle [F:\mathbb {Q} ]}$  is the degree of ${\displaystyle F}$  over ${\displaystyle \mathbb {Q} }$ .

Statement of the theorem

Stickelberger's Theorem^[4]
Let ${\displaystyle F}$  be an abelian number field. Then, the Stickelberger ideal of ${\displaystyle F}$  annihilates the class group of ${\displaystyle F}$ .

Note that ${\displaystyle \theta (F)}$  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 ${\displaystyle \alpha \in \mathbb {Z} [G_{F}]}$  is such that $${\displaystyle \alpha \theta (F)=\sum _{\sigma \in G_{F}}a_{\sigma }\sigma \in \mathbb {Z} [G_{F}]}$$  and if ${\displaystyle J}$  is any fractional ideal of ${\displaystyle F}$ , 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