Note on the class number of the $p$th cyclotomic field

Abstract

Let $p$ be a prime number of the form $p=2\ell^f+1$ with an odd prime number $\ell$, and $h_p^-$ the relative class number of the $p$th cyclotomic field $K=\mathbb{Q}(\zeta_p)$. When $f=1$, it is conjectured that $h_p^-$ is odd, and there are several results related to this conjecture. In this paper, we deal with the case $f \geq 2$. For $0 \leq t \leq f$, let $h_{p,t}^-$ denote the relative class number of the imaginary subfield $K_t$ of $K$ of degree $2\ell^t$ over $\mathbb{Q}$. We show that the ratio $h_{p,f}^-/h_{p,f-1}^-$ is not divisible by a prime number $r$ if $r$ is a primitive root modulo $\ell^2$. Further, when $r \leq 47$, we give some computational results on the ratio $h_{p,t}^-/h_{p,t-1}^-$ for $1 \leq t \leq f$. In the range of our computation, we find that the ratio is divisible by $r$ only in some exceptional cases.