On the exact divisibility by 5 of the class number of some pure metacyclic fields (Q6610527)

Consider a pure quintic field \(\Gamma = \mathbb{Q}(\sqrt[5]{n})\) with \(n\) fifth-power free. Let \(k = \Gamma(\zeta_5)\), the normal closure of \(\Gamma\). The respective \(5\)-ranks \(h_{\Gamma,5}\) and \(h_{k,5}\) of the class groups of \(\Gamma\) and \(k\) were studied by the reviewer, Kulkarni and Majumdar [\textit{M. Kulkarni} et al., J. Ramanujan Math. Soc. 30, No. 4, 413--454 (2015; Zbl 1425.11177)] using genus theory. The present authors had extended these results in an earlier where they posed a nice conjecture. The objective of the present paper is to prove this conjecture; with the notations as above, their main result asserts the following.\N\NLet \(q_1, q_2 \equiv \pm{2}\) mod \(5\) be primes such that \(q_2\) and \(5\) are not quintic residues mod \(q_1\). Then \(h_{\Gamma,5}=1\) and \(h_{k,5}=5\) in the following cases:\N\N\(n = 5q_1\) or \(q_1q_2\) or \(5q_1q_2\) respectively, according as to when \(q_1 \equiv \pm{7}\) mod \(25\) or when \(q_1,q_2 \equiv \pm{7}\) mod \(25\) or when at least one of \(q_1, q_2\) is \(\not\equiv \pm{7}\) mod \(25\).\N\NAmong other things, they also use the results due to the reviewer and others mentioned above.\N\NFor the entire collection see [Zbl 1539.11005].