Open Access
June 2015 Note on the class number of the $p$th cyclotomic field
Shoichi Fujima, Humio Ichimura
Funct. Approx. Comment. Math. 52(2): 299-309 (June 2015). DOI: 10.7169/facm/2015.52.2.8


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.


Download Citation

Shoichi Fujima. Humio Ichimura. "Note on the class number of the $p$th cyclotomic field." Funct. Approx. Comment. Math. 52 (2) 299 - 309, June 2015.


Published: June 2015
First available in Project Euclid: 18 June 2015

zbMATH: 06862264
MathSciNet: MR3358322
Digital Object Identifier: 10.7169/facm/2015.52.2.8

Primary: 11R18
Secondary: 11R29

Keywords: computation , cyclotomic field , relative class number

Rights: Copyright © 2015 Adam Mickiewicz University

Vol.52 • No. 2 • June 2015
Back to Top