Some applications of Kummer and Stickelberger relations
Abstract
Let p be an odd prime. Let K = Q(zeta) be the p-cyclotomic field. Let v be any primitive root mod p. Let sigma be a Q-isomorphism of K. Let P(sigma) = sigma^{p-2}v^{-(p-2)}+ ... + sigma v^{-1} +1 \in Z[G] where 1 \leq v^n \leq p-1 is a notation mod p. We apply a Kummer and Stickelberger relation of K to some singular not primary numbers A of K connected to p-class group C_p of K and prove they verify the congruence A^P(sigma) = 1 mod p^2. This p-adic method on singular numbers A allows us to prove: in a straightforward way the connection between relative p-class group C_p^- and the solutions of some explicit congruences mod p in Z[X]: \sum_{i=1}^{p-2} ((v^{-(i-1)} - v^{-i} v) /p) X^{i-1} \equiv 0 mod p and that if (p-1)/2 is odd then the Bernoulli Number B_((p+1)/2) not = 0 mod p. In this version some congruences deduced of Stickelberger relation for prime ideals Q of K of inertial degree f > 1 are added.
Cite
@article{arxiv.math/0601136,
title = {Some applications of Kummer and Stickelberger relations},
author = {Roland Queme},
journal= {arXiv preprint arXiv:math/0601136},
year = {2007}
}
Comments
The sufficient condition for equality of the rank of the relative p-class group C_p^- and the index of irregularity i_p of K added in previous version not correct is removed of this version. In this version some congruences deduced of Stickelberger relation for prime ideals Q of K of inertial degree f > 1 are added