## Functiones et Approximatio Commentarii Mathematici

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

Humio Ichimura

#### Abstract

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

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

Dates
First available in Project Euclid: 28 March 2017

https://projecteuclid.org/euclid.facm/1490688020

Digital Object Identifier
doi:10.7169/facm/1619

Mathematical Reviews number (MathSciNet)
MR3704228

Zentralblatt MATH identifier
06864166

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

#### Citation

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. https://projecteuclid.org/euclid.facm/1490688020

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