On the class number of cyclic extensions K/Q

Abstract

Let K/Q be a cyclic extension. In this paper, we give several congruences connecting the prime divisors of the degree g= [K:Q] with the prime divisors of the class number h of K/Q. As an exemple, the theorem: Let K/Q be a cyclic extension with [K:Q]=g. Suppose that g is not divisible by 2 . Let g_j, j=1,...m, be the prime divisors of g. Let h_i, i=1,...r, be the prime divisors of the class number h of K/Q. If for one prime factor h_i of h, the h_i-component G(h_i) of the class group G of K/Q is cyclic then: else h_i divides g, else h_i = 1 (mod g_j) for at least one prime divisor g_j of g. The results obtained are all in accordance with class number tables of Washington, Masley, Girtsmair, Schoof, Jeannin and number fields server megrez.math.u-bordeaux.fr The proofs are strictly elementary.