l-Class Groups of Cyclic Extensions of Prime Degree l

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 ischolar_mains 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, explicit results for l=5 and F=Q(e^2iπ/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(e^2iπ/5n, x^1/5). Using the results on the class groups of the fields of the form Q(e^2iπ/5, x^1/5), and using Kummer duality theory, we deduce results on the 5-class numbers of fields of the form Q(x^1/5).