{"title":"单原子亚原子组成","authors":"J. Harrington, L. Jones","doi":"10.2996/kmj44107","DOIUrl":null,"url":null,"abstract":"Let $m$ and $n$ be positive integers, and let $p$ be a prime. Let $T(x)=\\Phi_{p^m}\\left(\\Phi_{2^n}(x)\\right)$, where $\\Phi_k(x)$ is the cyclotomic polynomial of index $k$. In this article, we prove that $T(x)$ is irreducible over $\\mathbb Q$ and that \\[\\left\\{1,\\theta,\\theta^2,\\ldots,\\theta^{2^{n-1}p^{m-1}(p-1)-1}\\right\\}\\] is a basis for the ring of integers of $\\mathbb Q(\\theta)$, where $T(\\theta)=0$.","PeriodicalId":54747,"journal":{"name":"Kodai Mathematical Journal","volume":null,"pages":null},"PeriodicalIF":0.4000,"publicationDate":"2019-09-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"6","resultStr":"{\"title\":\"Monogenic Cyclotomic compositions\",\"authors\":\"J. Harrington, L. Jones\",\"doi\":\"10.2996/kmj44107\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"Let $m$ and $n$ be positive integers, and let $p$ be a prime. Let $T(x)=\\\\Phi_{p^m}\\\\left(\\\\Phi_{2^n}(x)\\\\right)$, where $\\\\Phi_k(x)$ is the cyclotomic polynomial of index $k$. In this article, we prove that $T(x)$ is irreducible over $\\\\mathbb Q$ and that \\\\[\\\\left\\\\{1,\\\\theta,\\\\theta^2,\\\\ldots,\\\\theta^{2^{n-1}p^{m-1}(p-1)-1}\\\\right\\\\}\\\\] is a basis for the ring of integers of $\\\\mathbb Q(\\\\theta)$, where $T(\\\\theta)=0$.\",\"PeriodicalId\":54747,\"journal\":{\"name\":\"Kodai Mathematical Journal\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":0.4000,\"publicationDate\":\"2019-09-08\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"6\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Kodai Mathematical Journal\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://doi.org/10.2996/kmj44107\",\"RegionNum\":4,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q4\",\"JCRName\":\"MATHEMATICS\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Kodai Mathematical Journal","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.2996/kmj44107","RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q4","JCRName":"MATHEMATICS","Score":null,"Total":0}
Let $m$ and $n$ be positive integers, and let $p$ be a prime. Let $T(x)=\Phi_{p^m}\left(\Phi_{2^n}(x)\right)$, where $\Phi_k(x)$ is the cyclotomic polynomial of index $k$. In this article, we prove that $T(x)$ is irreducible over $\mathbb Q$ and that \[\left\{1,\theta,\theta^2,\ldots,\theta^{2^{n-1}p^{m-1}(p-1)-1}\right\}\] is a basis for the ring of integers of $\mathbb Q(\theta)$, where $T(\theta)=0$.
期刊介绍:
Kodai Mathematical Journal is edited by the Department of Mathematics, Tokyo Institute of Technology. The journal was issued from 1949 until 1977 as Kodai Mathematical Seminar Reports, and was renewed in 1978 under the present name. The journal is published three times yearly and includes original papers in mathematics.