On the growth of the (S,{2})-refined class number

Abstract

We give a new proof of the fact that the growth (with respect to n) of the p-adic valuation of hn,2− is linear, where hn,2− denotes the minus part of the (S,{2})-refined class number of the cyclotomic field Q(μpn+1), as defined by Hu and Kim in [1]. As a consequence of our proof, we obtain an explicit relation between the p-adic valuation of hn,2− and the p-adic valuation of hn−, the minus part of the class number hn of the cyclotomic field Q(μpn+1).