Proceedings of the Japan Academy, Series A, Mathematical Sciences

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

Humio Ichimura and Shoichi Nakajima

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Abstract

Let $p$ be a prime number with $p \equiv 3 \bmod 4$ and $q=(p-1)/2$. Let $k=\mathbf{Q}(\sqrt{-p})$ and $k_{\infty}/k$ be the cyclotomic $\mathbf{Z}_{p}$-extension. Denote by $h_{n}^{-}$ the relative class number of the $n$-th layer $k_{n}$. Let $\ell$ be a prime number with $\ell \neq p$. We show that, for any $n \geq 1$, $\ell$ does not divide $h_{n}^{-}/h_{n-1}^{-}$ (resp. $h_{1}^{-}/h_{0}^{-}$) if $\ell$ is a primitive root modulo $p^{2}$ (resp. $p$) and $\ell \geq q-2$ (resp. $\ell \geq q-6$). Further, we show with the help of computer that when $p < 10000$ and $n \leq 100$, $\ell$ does not divide $h_{n}^{-}/h_{n-1}^{-}$ (resp. $h_{1}^{-}/h_{0}^{-}$) for any prime $\ell$ which is a primitive root modulo $p^{2}$ (resp. $p$).

Article information

Source
Proc. Japan Acad. Ser. A Math. Sci., Volume 88, Number 1 (2012), 16-20.

Dates
First available in Project Euclid: 30 December 2011

Permanent link to this document
https://projecteuclid.org/euclid.pja/1325264391

Digital Object Identifier
doi:10.3792/pjaa.88.16

Mathematical Reviews number (MathSciNet)
MR2872210

Zentralblatt MATH identifier
1333.11103

Subjects
Primary: 11R18: Cyclotomic extensions

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

Citation

Ichimura, Humio; Nakajima, Shoichi. A note on the relative class number of the cyclotomic $\mathbf{Z}_{p}$-extension of $\mathbf{Q}(\sqrt{-p})$. Proc. Japan Acad. Ser. A Math. Sci. 88 (2012), no. 1, 16--20. doi:10.3792/pjaa.88.16. https://projecteuclid.org/euclid.pja/1325264391


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