Proceedings of the Japan Academy, Series A, Mathematical Sciences

A note on the relative class number of the cyclotomic $\mathbf{Z}_{p}$-extension of $\mathbf{Q}(\sqrt{-p})$, II

Humio Ichimura

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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 $p^{2}$.

Article information

Source
Proc. Japan Acad. Ser. A Math. Sci., Volume 89, Number 2 (2013), 21-23.

Dates
First available in Project Euclid: 30 January 2013

Permanent link to this document
https://projecteuclid.org/euclid.pja/1359554916

Digital Object Identifier
doi:10.3792/pjaa.89.21

Mathematical Reviews number (MathSciNet)
MR3024270

Zentralblatt MATH identifier
1334.11084

Subjects
Primary: 11R18: Cyclotomic extensions

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

Citation

Ichimura, Humio. 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 (2013), no. 2, 21--23. doi:10.3792/pjaa.89.21. https://projecteuclid.org/euclid.pja/1359554916


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