On class numbers inside the real $p$th cyclotomic field

Abstract

We fix an integer $n \geq 1$, a prime number $\ell$ with $\ell \not\mid 2n$ and an integer $s \geq 0$. We deal with a prime number $p$ of the form $p=2n\ell^f+1$. For $0 \leq t \leq f$, let $K_t$ be the real cyclic field of degree $\ell^t$ contained in the $p$th cyclotomic field, and let $h_t$ be the class number of $K_t$. We show that when $p$ (or $f$) is large enough with respect to $n$, $\ell$ and $s$, a prime number $r$ does not divide the ratio $h_f/h_{f-(s+1)}$ whenever $r$ is a primitive root modulo $\ell^2$.