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

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 $\mathbb{Q}(\mu_{p^{n+1}})$, as defined by Hu and Kim in [J. of Number Theory 158 (2016), 73-89]. 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 $\mathbb{Q}(\mu_{p^{n+1}})$.