2006/01/07 by Roland Quême, Roland Queme, Queme, Roland
Computer Science · Mathematics · #Advanced Algebra and Logic #Advanced Combinatorial Mathematics #Functional Equations Stability Results #math.NT #msc:11R18 #msc:11R29
paper · pdf · doi:10.48550/arxiv.math/0601136
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
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) = sigmap-2v-(p-2)+ ... + sigma v-1 +1 ∈ Z[G] where 1 ≤ vn ≤ 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 Cp of K and prove they verify the congruence AP(sigma) = 1 mod p2. This p-adic method on singular numbers A allows us to prove: in a straightforward way the connection between relative p-class group Cp- and the solutions of some explicit congruences mod p in Z[X]: ∑i=1p-2 ((v-(i-1) - v-i v) /p) Xi-1 ≡ 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.