{"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.
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。
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.