Relative class numbers inside the $p$th cyclotomic field

IF 0.5 4区 数学 Q3 MATHEMATICS
H. Ichimura
{"title":"Relative class numbers inside the $p$th cyclotomic field","authors":"H. Ichimura","doi":"10.18910/77238","DOIUrl":null,"url":null,"abstract":"For a prime number p ≡ 3 mod 4, we write p = 2 n (cid:2) f + 1 for some power (cid:2) f of an odd prime number (cid:2) and an odd integer n with (cid:2) (cid:2) n . For 0 ≤ t ≤ f , let K t be the imaginary subfield of Q ( ζ p ) of degree 2 (cid:2) t and let h − t be the relative class number of K t . We show that for n = 1 (resp. n ≥ 3), a prime number r does not divide the ratio h − t / h − t − 1 when r is a primitive root modulo (cid:2) 2 and r ≥ (cid:2) f − t − 1 (resp. r ≥ ( n − 2) (cid:2) f − t + 1). In particular, for n = 1 or 3, the ratio h − f / h − f − 1 at the top is not divisible by r whenever r is a primitive root modulo (cid:2) 2 . Further, we show that the (cid:2) -part of h − t / h − t − 1 stabilizes for “large” t under some assumption.","PeriodicalId":54660,"journal":{"name":"Osaka Journal of Mathematics","volume":"57 1","pages":"949-959"},"PeriodicalIF":0.5000,"publicationDate":"2020-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"1","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Osaka Journal of Mathematics","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.18910/77238","RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q3","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 1

Abstract

For a prime number p ≡ 3 mod 4, we write p = 2 n (cid:2) f + 1 for some power (cid:2) f of an odd prime number (cid:2) and an odd integer n with (cid:2) (cid:2) n . For 0 ≤ t ≤ f , let K t be the imaginary subfield of Q ( ζ p ) of degree 2 (cid:2) t and let h − t be the relative class number of K t . We show that for n = 1 (resp. n ≥ 3), a prime number r does not divide the ratio h − t / h − t − 1 when r is a primitive root modulo (cid:2) 2 and r ≥ (cid:2) f − t − 1 (resp. r ≥ ( n − 2) (cid:2) f − t + 1). In particular, for n = 1 or 3, the ratio h − f / h − f − 1 at the top is not divisible by r whenever r is a primitive root modulo (cid:2) 2 . Further, we show that the (cid:2) -part of h − t / h − t − 1 stabilizes for “large” t under some assumption.
第$p$th个分圆域内的相对类数
对于素数p lect 3 mod 4,我们为奇数素数(cid:2)和奇数整数n的某个幂(cid:2)f写p=2n(cid:2)f+1。对于0≤t≤f,设KT为2(cid:2)t阶Q(ζp)的虚子域,设h−t为KT的相对类数。我们证明,对于n=1(分别为n≥3),当r是基根模(cid:2)2且r≥(cid:2)f−t−1时,素数r不除以比率h−t/h−t−1。特别是,对于n=1或3,当r是模(cid:2)2的原始根时,顶部的比率h−f/h−f−1不可被r整除。此外,我们证明了在某种假设下,h−t/h−t−1的(cid:2)-部分对“大”t稳定。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
求助全文
约1分钟内获得全文 求助全文
来源期刊
CiteScore
0.90
自引率
0.00%
发文量
0
审稿时长
>12 weeks
期刊介绍: Osaka Journal of Mathematics is published quarterly by the joint editorship of the Department of Mathematics, Graduate School of Science, Osaka University, and the Department of Mathematics, Faculty of Science, Osaka City University and the Department of Pure and Applied Mathematics, Graduate School of Information Science and Technology, Osaka University with the cooperation of the Department of Mathematical Sciences, Faculty of Engineering Science, Osaka University. The Journal is devoted entirely to the publication of original works in pure and applied mathematics.
×
引用
GB/T 7714-2015
复制
MLA
复制
APA
复制
导出至
BibTeX EndNote RefMan NoteFirst NoteExpress
×
提示
您的信息不完整,为了账户安全,请先补充。
现在去补充
×
提示
您因"违规操作"
具体请查看互助需知
我知道了
×
提示
确定
请完成安全验证×
copy
已复制链接
快去分享给好友吧!
我知道了
右上角分享
点击右上角分享
0
联系我们:info@booksci.cn Book学术提供免费学术资源搜索服务,方便国内外学者检索中英文文献。致力于提供最便捷和优质的服务体验。 Copyright © 2023 布克学术 All rights reserved.
京ICP备2023020795号-1
ghs 京公网安备 11010802042870号
Book学术文献互助
Book学术文献互助群
群 号:481959085
Book学术官方微信