On Kummer and Stickelberger relations

Abstract

Let p be an odd prime. Let K_p = \Q(zeta_p) be the p-cyclotomic field. We apply a Kummer and Stickelberger relation of K_p to some singular not primary numbers A of K_p connected to p-class group of K_p and prove they verify the congruence A = 1 mod p^2. Let v be a primitive root mod p. This p-adic improvement on singular numbers A allows us to connect in a straightforward way the p-class group C_p to the solutions of some explicit congruence mod p: \sum_{i=1}^{p-2} X^{i-1} \times (\frac{v^{-(i-1)}-v^{-i}\times v}{p}) \equiv 0 mod p: where X is a natural integer and where v^n is understood as v^n mod p with 1 \leq v^n \leq p-1 with n integer \in \Z. The numerical verification of this congruence is completely consistent with table of irregular primes in Washington p. 410.