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. It is due to Ludwig Stickelberger (1890).

Theorem (Stickelberger)Let $mathbb{Q}(zeta_m)$ be a cyclotomic field extension of $mathbb{Q}$ with Galois group $G = {sigma_a | a in (mathbb Z / mmathbb Z)^*}$ , and consider the group ring $mathbb{Q} [G]$ . Define the Stickelberger element $heta in mathbb{Q} [G]$ by : $heta = frac 1 m sum_{a in (mathbb Z / mmathbb Z)^*} a sigma_a^{-1}$

and take $eta in mathbb{Z} [G]$ such that $eta heta in mathbb{Z} [G]$ as well. Then $eta hetamathbb{}$ is an annihilator for the ideal class group of $mathbb{Q}(zeta_m)$ , as Galois module.

Note that $heta$ itself need not be an annihilator, just that any multiple of it in $mathbb{Z} [G]$ is.

References

* Ludwig Stickelberger, [http://gdz.sub.uni-goettingen.de/no_cache/dms/load/img/?IDDOC=27547 "Ueber eine Verallgemeinerung der Kreistheilung"] , Mathematische Annalen 37 (1890), S. 321–367
*Boas Erez, [http://www.fen.bilkent.edu.tr/~franz/publ/boas.pdf "Darstellungen von Gruppen in der Algebraischen Zahlentheorie: eine Einführung"]

External links

* [http://planetmath.org/?op=getobj&from=objects&id=5642 PlanetMath page]

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