l-Class groups of cyclic extensions of prime degree l

TL;DR

This paper investigates the structure and bounds of the l-Sylow subgroup of class groups in cyclic prime degree extensions, providing explicit results especially for l=5 and applications to fields like Q(n^{1/5}).

Contribution

It introduces new bounds and explicit formulas for the l-class groups of cyclic extensions, especially for l=5, using genus theory and residue symbols.

Findings

Bounds for the l-rank of class groups in cyclic extensions.

Explicit formulas for l=5 class groups using residue symbols.

Results on 5-class numbers of fields Q(n^{1/5}) derived.

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_ 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(\zeta_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).