Let K/F be a cyclic extension of odd 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 Fl -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, explicit results for l = 5 and F = ℚ(e2iπ/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. These results have a number of potential applications. For instance, some of them like Theorem 5.16 could be useful in the arithmetic of elliptic curves over towers of the form ℚ(e2iπ/5n, x1/5). Using the results on the class groups of the fields of the form ℚ(e2iπ/5, x1/5), and using Kummer duality theory, we deduce results on the 5-class numbers of fields of the form ℚ(x1/5). © 2015 Ramanujan Mathematical Society.