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 h n , 2 − is linear, where h n , 2 − denotes the minus part of the (S , { 2 }) -refined class number of the cyclotomic field Q (μ p n + 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 h n , 2 − and the p -adic valuation of h n − , the minus part of the class number h n of the cyclotomic field Q (μ p n + 1 ). [ABSTRACT FROM AUTHOR]