{"title":"论由 $$x^9+ax^5+b$$ 定义的某些 Nonic 数域的索引","authors":"Omar Kchit","doi":"10.1134/s2070046624020018","DOIUrl":null,"url":null,"abstract":"<h3 data-test=\"abstract-sub-heading\">Abstract</h3><p> In this paper, for any nonic number field <span>\\(K\\)</span> generated by a root <span>\\(\\alpha\\)</span> of a monic irreducible trinomial <span>\\(F(x)=x^9+ax^5+b \\in \\mathbb{Z}[x]\\)</span> and for every rational prime <span>\\(p\\)</span>, we characterize when <span>\\(p\\)</span> divides the index of <span>\\(K\\)</span>. We also describe the prime power decomposition of the index <span>\\(i(K)\\)</span>. In such a way we give a partial answer of Problem <span>\\(22\\)</span> of Narkiewicz [23] for this family of number fields. As an application of our results, if <span>\\(i(K)\\neq1\\)</span>, then <span>\\(K\\)</span> is not monogenic. We illustrate our results by some computational examples. </p>","PeriodicalId":44654,"journal":{"name":"P-Adic Numbers Ultrametric Analysis and Applications","volume":"32 1","pages":""},"PeriodicalIF":0.5000,"publicationDate":"2024-05-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"On the Index of Certain Nonic Number Fields Defined by $$x^9+ax^5+b$$\",\"authors\":\"Omar Kchit\",\"doi\":\"10.1134/s2070046624020018\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<h3 data-test=\\\"abstract-sub-heading\\\">Abstract</h3><p> In this paper, for any nonic number field <span>\\\\(K\\\\)</span> generated by a root <span>\\\\(\\\\alpha\\\\)</span> of a monic irreducible trinomial <span>\\\\(F(x)=x^9+ax^5+b \\\\in \\\\mathbb{Z}[x]\\\\)</span> and for every rational prime <span>\\\\(p\\\\)</span>, we characterize when <span>\\\\(p\\\\)</span> divides the index of <span>\\\\(K\\\\)</span>. We also describe the prime power decomposition of the index <span>\\\\(i(K)\\\\)</span>. In such a way we give a partial answer of Problem <span>\\\\(22\\\\)</span> of Narkiewicz [23] for this family of number fields. As an application of our results, if <span>\\\\(i(K)\\\\neq1\\\\)</span>, then <span>\\\\(K\\\\)</span> is not monogenic. We illustrate our results by some computational examples. </p>\",\"PeriodicalId\":44654,\"journal\":{\"name\":\"P-Adic Numbers Ultrametric Analysis and Applications\",\"volume\":\"32 1\",\"pages\":\"\"},\"PeriodicalIF\":0.5000,\"publicationDate\":\"2024-05-06\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"P-Adic Numbers Ultrametric Analysis and Applications\",\"FirstCategoryId\":\"1085\",\"ListUrlMain\":\"https://doi.org/10.1134/s2070046624020018\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q4\",\"JCRName\":\"MATHEMATICS, INTERDISCIPLINARY APPLICATIONS\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"P-Adic Numbers Ultrametric Analysis and Applications","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.1134/s2070046624020018","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q4","JCRName":"MATHEMATICS, INTERDISCIPLINARY APPLICATIONS","Score":null,"Total":0}
On the Index of Certain Nonic Number Fields Defined by $$x^9+ax^5+b$$
Abstract
In this paper, for any nonic number field \(K\) generated by a root \(\alpha\) of a monic irreducible trinomial \(F(x)=x^9+ax^5+b \in \mathbb{Z}[x]\) and for every rational prime \(p\), we characterize when \(p\) divides the index of \(K\). We also describe the prime power decomposition of the index \(i(K)\). In such a way we give a partial answer of Problem \(22\) of Narkiewicz [23] for this family of number fields. As an application of our results, if \(i(K)\neq1\), then \(K\) is not monogenic. We illustrate our results by some computational examples.
期刊介绍:
This is a new international interdisciplinary journal which contains original articles, short communications, and reviews on progress in various areas of pure and applied mathematics related with p-adic, adelic and ultrametric methods, including: mathematical physics, quantum theory, string theory, cosmology, nanoscience, life sciences; mathematical analysis, number theory, algebraic geometry, non-Archimedean and non-commutative geometry, theory of finite fields and rings, representation theory, functional analysis and graph theory; classical and quantum information, computer science, cryptography, image analysis, cognitive models, neural networks and bioinformatics; complex systems, dynamical systems, stochastic processes, hierarchy structures, modeling, control theory, economics and sociology; mesoscopic and nano systems, disordered and chaotic systems, spin glasses, macromolecules, molecular dynamics, biopolymers, genomics and biology; and other related fields.