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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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

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

First available in Project Euclid: 18 June 2015

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

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

relative class number cyclotomic field computation


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.

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  • P.T. Bateman and R. A. Horn, A heuristic asymptotic formula concerning the distribution of prime numbers, Math. Comp. 16 (1962), 363–367.
  • P.E. Conner and J. Hurrelbrink, Class Number Parity, World Scientific, Singapore, 1988.
  • D. Davis, Computing the number of totally positive circular units which are square, J. Number Theory 10 (1978), 1–9.
  • D.R. Estes, On the parity of the class number of the field of $q$-th roots of unity, Rocky Mountain J. Math. 19 (1989), 675–682.
  • H. Ichimura and S. Nakajima, On the $2$-part of the class numbers of cyclotomic fields of prime power conductors, J. Math. Soc. Japan 64 (2012), 317–342.
  • H. Ichimura and S. Nakajima, A note on the relative class number of the cyclotomic $\ADHBZ_p$-extension of $\ADHBQ(\sqrt{-p})$, Proc. Japan Acad. 88A (2012), 16–20.
  • H. Ichimura, S. Nakajima and H. Sumida-Takahashi, On the Iwasawa lambda invariant of an imaginary abelian field of conductor $3p^{n+1}$, J. Number Theory 133 (2013), 787–801.
  • S. Jakubec, On the class number of some real abelian fields of prime conductors, Acta Arith. 145 (2010), 315–318.
  • T. Metsänkylä, Some divisibility results for the cyclotomic class number, Tatra Mt. Math. Publ. 11 (1997), 59-68.
  • T. Metsänkylä, An application of the $p$-adic class number formula, Manuscripta Math. 93 (1997), 481–498.
  • P. Stevenhagen, Class number parity of the $p$th cyclotomic field, Math. Comp. 63 (1994), 773–784.
  • L.C. Washington, The non-$p$-part of the class number in cyclotomic $\ADHBZ_p$-extension, Invent. Math. 49 (1979), 87–97.
  • L.C. Washington, Introduction to Cyclotomic Fields (2nd ed.), Springer, New York, 1997.
  • K. Yamamura,
  • K. Yoshino, A condition for divisibility of the class number of real $p$th cyclotomic field by an odd prime number distinct from $p$, Abh. Math. Semin. Hamburg 69 (1999), 37–57.