l-Class groups of cyclic extensions of prime degree l

Abstract

Let K/F be a cyclic extension of prime degree l over a number field F . If F has class number coprime to l, we study the structure of the l-Sylow subgroup of the class group of K. In particular, when F contains the l-th roots of unity, we obtain bounds for the F l -rank of the l-Sylow subgroup of K using genus theory. We obtain some results valid for general l. Following that, we obtain more complete results for l = 5 and F = Q(ζ 5 ). The rank of the 5-class group of K is expressed in terms of power residue symbols. We compare our results with tables obtained using SAGE (the latter is under GRH). We obtain explicit results in several cases. Using these results, and duality theory, we deduce results on the 5-class numbers of fields of the form Q(n 1/5 ).