On prime factors of class number of cyclotomic fields

Roland Queme

Published 2006-09-26, updated 2007-01-10Version 5

Let p be an odd prime. Let K = \Q(zeta) be the p-cyclotomic number field. Let v be a primitive root mod p and sigma : zeta --> zeta^v be a \Q-isomorphism of the extension K/\Q generating the Galois group G of K/\Q. For n in Z, the notation v^n is understood by v^n mod p with 1 \leq v^n \leq p-1. Let P(X) = \sum_{i=0}^{p-2} v^{-i}X^i \in \Z[X] be the Stickelberger polynomial. P(sigma) annihilates the class group C of K. There exists a polynomial Q(X) \in \Z[X] such that P(sigma)(sigma-v) = p\times Q(sigma) and such that Q(sigma) annihilates the p-class group C_p of K (the subgroup of exponent p of C). In the other hand sigma^{(p-1)/2}+1 annihilates the relative class group of K. The simultaneous application of these results brings some informations on the structure of the class group C, give some explicit congruences in \Z[v] mod p for the p-class group C_p of K and some explicit congruences in \Z[v] mod h for the h-class group of K for all the prime divisors h \not = p of the class number h(K). We detail at the end the case of class number of quadratic and biquadratic fields contained in the cyclotomic field K and give a general MAPLE algorithm.