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February 2013 A note on the relative class number of the cyclotomic $\mathbf{Z}_{p}$-extension of $\mathbf{Q}(\sqrt{-p})$, II
Humio Ichimura
Proc. Japan Acad. Ser. A Math. Sci. 89(2): 21-23 (February 2013). DOI: 10.3792/pjaa.89.21

Abstract

Let $p$ be a prime number with $p \equiv 3\,\mathrm{mod}\,4$, and let $k=\mathbf{Q}(\sqrt{-p})$. Denote by $h_{n}^{-}$ the relative class number of the $n$th layer of the cyclotomic $\mathbf{Z}_{p}$-extension over $k$. Let $q=(p-1)/2$ and $d_{p}$ be the largest divisor of $q$ with $d_{p} < q$. Let $\ell$ be a prime number with $\ell \neq p$. We show that $\ell \nmid h_{n}^{-}$ for all $n \geq 0$ if $\ell \geq q-2d_{p}$ and $\ell$ is a primitive root modulo $p^{2}$.

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Humio Ichimura. "A note on the relative class number of the cyclotomic $\mathbf{Z}_{p}$-extension of $\mathbf{Q}(\sqrt{-p})$, II." Proc. Japan Acad. Ser. A Math. Sci. 89 (2) 21 - 23, February 2013. https://doi.org/10.3792/pjaa.89.21

Information

Published: February 2013
First available in Project Euclid: 30 January 2013

zbMATH: 1334.11084
MathSciNet: MR3024270
Digital Object Identifier: 10.3792/pjaa.89.21

Subjects:
Primary: 11R18

Keywords: Class number , cyclotomic $\mathbf{Z}_{p}$-extension , non-$p$-part , quadratic field

Rights: Copyright © 2013 The Japan Academy

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Vol.89 • No. 2 • February 2013
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