Functiones et Approximatio Commentarii Mathematici

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

Humio Ichimura

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text


Let $p=2\ell^f+1$ be a prime number with $f \geq 2$ and an odd prime number $\ell$. For $0 \leq t \leq f$, let $K_t$ be the imaginary subfield of the $p$th cyclotomic field $\mathbb{Q}(\zeta_p)$ with $[K_t : \mathbb{Q}]=2\ell^t$. Denote by $h_{p,t}^-$ the relative class number of $K_t$, and by $h_{p,t}^+$ the class number of the maximal real subfield $K_t^+$. It is known that the ratio $h_{p,f}^-/h_{p,f-1}^-$ is odd (and hence so is $h_{p,f}^+/h_{p,f-1}^+$) whenever $2$ is a primitive root modulo $\ell^2$. We show that $h_{p,f}^+/h_{p,f-1}^+$ is odd under a somewhat milder assumption on $\ell$ and that the ratio $h_{p,f-1}^-/h_{p,f-2}^-$ is always odd when $\ell=3$.

Article information

Funct. Approx. Comment. Math., Volume 57, Number 1 (2017), 93-103.

First available in Project Euclid: 28 March 2017

Permanent link to this document

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


Ichimura, Humio. Note on the class number of the $p$th cyclotomic field, III. Funct. Approx. Comment. Math. 57 (2017), no. 1, 93--103. doi:10.7169/facm/1619.

Export citation


  • P. Cornacchia, The parity of class number of the cyclotomic fields of prime conductor, Proc. Amer. Math. Soc., 125 (1997), 3163-3168.
  • P. Cornacchia and C. Greither, Fitting ideals of class groups of real fields of prime power conductor, J. Number Theory, 73 (1998), 459-471.
  • D. Estes, On the parity of the class number of the field of $q$ th roots of unity, Rocky Mountain J. Math., 19 (1989), 675-689.
  • S. Fujima and H. Ichimura, Note on the class number of the $p$th cyclotomic field, Funct. Approx. Comment. Math., 52.2 (2015), 299-309.
  • S. Fujima and H. Ichimura, Note on the class number of the $p$th cyclotomic field, II, Experiment. Math., doi:10.1080/10586458.2016.1230528.
  • C. Greither, Class groups of abelian extensions and the main conjecture, Ann. Inst. Fourier, 42 (1996), 449-499.
  • K. Horie, Ideal class groups of the Iwasawa-theoretical extensions of the rationals, J. London Math. Soc., 66 (2002), 257-275.
  • H. Ichimura, On a duality of Gras between totally positive and primary cyclotomic units, Math. J. Okayama Univ., 58 (2016), 125-132.
  • S. Jakubec and P. Trojovský, On divisiblity of the class number $h^+$ of the real cyclotomic fields $\AFAIQ(\zeta_p+\zeta_p^{-1})$ by primes $p < 5000$, Abh. Math. Univ. Hamburg, 67 (1997), 269-280.
  • T. Metsänkylä, Some divisibility results for the cyclotomic class numbers, Tatra Mt. Math. Publ., 11 (1997), 59-68.
  • W. Narkiewicz, Elementary and Analytic Theory of Algebraic Numbers (3rd. ed.), Springer, Berlin, 2004.
  • P. Stevenhagen, Class number parity of the $p$th cyclotomic field, Math. Comp., 63 (1994), 773-784.
  • P. Trojovský, On divisibility of the class number $h^+$ of the real cyclotomic field $\AFAIQ(\zeta_q+\zeta_q^{-1})$ by primes $q < 10000$, Math. Slovaca, 50.5 (2000), 541-555.
  • L.C. Washington, Introduction to Cyclotomic Fields (2nd ed.), Springer, New York, 1997.
  • H.C. Williams and C.R. Zarnke, Some prime numbers of the forms $2A3^n+1$ and $2A3^n-1$, Math. Comp., 26 (1972), 995-998.