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Stickelberger's theorem

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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

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Let Km denote the mth cyclotomic field, i.e. the extension of the rational numbers obtained by adjoining the mth roots of unity to (where m ≥ 2 is an integer). It is a Galois extension of with Galois group Gm isomorphic to the multiplicative group of integers modulo m (/m)×. The Stickelberger element (of level m or of Km) is an element in the group ring [Gm] and the Stickelberger ideal (of level m or of Km) is an ideal in the group ring [Gm]. They are defined as follows. Let ζm denote a primitive mth root of unity. The isomorphism from (/m)× to Gm is given by sending a to σa defined by the relation

.

The Stickelberger element of level m is defined as

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

More generally, if F be any Abelian number field whose Galois group over is denoted GF, 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 Km. Fix the least such m (this is the (finite part of the) conductor of F over ). There is a natural group homomorphism GmGF given by restriction, i.e. if σGm, its image in GF is its restriction to F denoted resmσ. The Stickelberger element of F is then defined as

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

In the special case where F = Km, the Stickelberger ideal I(Km) is generated by (aσa)θ(Km) as a varies over /m. This not true for general F.[2]

Examples

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

where φ is the Euler totient function and [F : ] is the degree of F over .

Statement of the theorem

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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 [GF] is.

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

and if J is any fractional ideal of F, then

is a principal ideal.

See also

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Notes

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  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

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  • 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
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