Functiones et Approximatio Commentarii Mathematici

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

Shoichi Fujima and Humio Ichimura

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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.

Article information

Source
Funct. Approx. Comment. Math., Volume 52, Number 2 (2015), 299-309.

Dates
First available in Project Euclid: 18 June 2015

Permanent link to this document
https://projecteuclid.org/euclid.facm/1434650883

Digital Object Identifier
doi:10.7169/facm/2015.52.2.8

Mathematical Reviews number (MathSciNet)
MR3358322

Zentralblatt MATH identifier
06862264

Subjects
Primary: 11R18: Cyclotomic extensions
Secondary: 11R29: Class numbers, class groups, discriminants

Keywords
relative class number cyclotomic field computation

Citation

Fujima, Shoichi; Ichimura, Humio. Note on the class number of the $p$th cyclotomic field. Funct. Approx. Comment. Math. 52 (2015), no. 2, 299--309. doi:10.7169/facm/2015.52.2.8. https://projecteuclid.org/euclid.facm/1434650883


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