  We disprove, by means of numerical examples, the existence of a
Riemann-Hurwitz formula for the p-ranks of relative class groups in a
p-ramified p-extension K/k of number fields of CM-type containing ?\_p. In the
cyclic case of degree p, under some assumptions on the p-class group of k, we
prove some properties of the Galois structure of the p-class group of K; but we
have found, through numerical experimentation, that some theoretical group
structures do not exist in this particular situation, and we justify this fact.
Then we show, in this context, that Kida's formula on lambda invariants is
valid for the p-ranks if and only if the p-class group of K is reduced to the
group of ambiguous classes, which is of course not always the case.
