  This paper is an updated version of ANT-0372 (2002 dec 4) with the same
title. Several errors are corrected in this version.
  An example of the kind of results obtained is: Let K/\Q be an abelian
extension with N = [K:\Q] > 1, N odd. Let h(K) be the class number of K.
Suppose that h(K) > 1. Let p be a prime dividing h(K). Let r_p be the rank of
the p-class group of K. Then p \times (p^{r_p}-1) and N are not coprime.
  The paper is at elementary level and contains a lot of numerical examples.
