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September 2017 Note on the class number of the $p$th cyclotomic field, III
Humio Ichimura
Funct. Approx. Comment. Math. 57(1): 93-103 (September 2017). DOI: 10.7169/facm/1619

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 $\ell=3$.

Citation

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Humio Ichimura. "Note on the class number of the $p$th cyclotomic field, III." Funct. Approx. Comment. Math. 57 (1) 93 - 103, September 2017. https://doi.org/10.7169/facm/1619

Information

Published: September 2017
First available in Project Euclid: 28 March 2017

zbMATH: 06864166
MathSciNet: MR3704228
Digital Object Identifier: 10.7169/facm/1619

Subjects:
Primary: 11R18
Secondary: 11R29

Keywords: cyclotomic field , relative class number

Rights: Copyright © 2017 Adam Mickiewicz University

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Vol.57 • No. 1 • September 2017
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