|
|
|
|
Stickelberger's theorem
|
(Theorem)
|
|
Theorem 1 (Stickelberger) Let $L=\mathbb{Q}(\zeta_m)$ be a cyclotomic field extension of $\Q$ with Galois group $G=\{\sigma_a\}_{a\in(\Z/m\Z)^\times}$ , and consider the group ring $\Q[G]$ . Define the Stickelberger element $\theta\in\Q[G]$ by
and take $\beta\in\Z[G]$ such that $\beta\theta\in\Z[G]$ as well. Then $\beta\theta$ is an annihilator for the ideal class group of $\Q(\zeta_m)$ .
Note that $\theta$ itself need not be an annihilator, just that any multiple of it in $\Z[G]$ is.
This result allows for the most basic connections between the (otherwise hard to determine) structure of a cyclotomic ideal class group and its collection of annihilators. For an application of Stickelberger's theorem, see Herbrand's theorem.
|
"Stickelberger's theorem" is owned by mathcam.
|
|
(view preamble | get metadata)
| Also defines: |
Stickelberger element |
|
|
Cross-references: Herbrand's theorem, application, multiple, ideal class group, annihilator, group ring, Galois group, extension, cyclotomic field
There are 2 references to this entry.
This is version 3 of Stickelberger's theorem, born on 2004-02-27, modified 2004-03-02.
Object id is 5642, canonical name is StickelbergersTheorem.
Accessed 3417 times total.
Classification:
| AMS MSC: | 11R29 (Number theory :: Algebraic number theory: global fields :: Class numbers, class groups, discriminants) |
|
|
|
|
|
|
Pending Errata and Addenda
|
|
|
|
|
|
|
|
|
|
|