{"title":"关于由x^{84}-m定义的纯数域的幂积分基","authors":"Lhoussain El Fadil, Omar Kchit, Hanan Choulli","doi":"10.33044/revuma.2836","DOIUrl":null,"url":null,"abstract":"Let $K$ be a pure number field generated by a complex root of a monic irreducible polynomial $F(x)=x^{60}-m\\in \\mathbb{Z}[x]$, with $m\\neq \\pm1$ a square free integer. In this paper, we study the monogeneity of $K$. We prove that if $m\\not\\equiv 1\\md{4}$, $m\\not\\equiv \\mp 1 \\md{9} $ and $\\overline{m}\\not\\in\\{\\mp 1,\\mp 7\\} \\md{25}$, then $K$ is monogenic. But if $m\\equiv 1\\md{4}$, $m\\equiv \\mp1 \\md{9}$, or $m\\equiv \\mp 1\\md{25}$, then $K$ is not monogenic. Our results are illustrated by examples.","PeriodicalId":54469,"journal":{"name":"Revista De La Union Matematica Argentina","volume":"1 1","pages":"0"},"PeriodicalIF":0.6000,"publicationDate":"2023-06-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"On power integral bases of certain pure number fields defined by $x^{84}-m$\",\"authors\":\"Lhoussain El Fadil, Omar Kchit, Hanan Choulli\",\"doi\":\"10.33044/revuma.2836\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"Let $K$ be a pure number field generated by a complex root of a monic irreducible polynomial $F(x)=x^{60}-m\\\\in \\\\mathbb{Z}[x]$, with $m\\\\neq \\\\pm1$ a square free integer. In this paper, we study the monogeneity of $K$. We prove that if $m\\\\not\\\\equiv 1\\\\md{4}$, $m\\\\not\\\\equiv \\\\mp 1 \\\\md{9} $ and $\\\\overline{m}\\\\not\\\\in\\\\{\\\\mp 1,\\\\mp 7\\\\} \\\\md{25}$, then $K$ is monogenic. But if $m\\\\equiv 1\\\\md{4}$, $m\\\\equiv \\\\mp1 \\\\md{9}$, or $m\\\\equiv \\\\mp 1\\\\md{25}$, then $K$ is not monogenic. Our results are illustrated by examples.\",\"PeriodicalId\":54469,\"journal\":{\"name\":\"Revista De La Union Matematica Argentina\",\"volume\":\"1 1\",\"pages\":\"0\"},\"PeriodicalIF\":0.6000,\"publicationDate\":\"2023-06-28\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Revista De La Union Matematica Argentina\",\"FirstCategoryId\":\"1085\",\"ListUrlMain\":\"https://doi.org/10.33044/revuma.2836\",\"RegionNum\":4,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q3\",\"JCRName\":\"MATHEMATICS\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Revista De La Union Matematica Argentina","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.33044/revuma.2836","RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q3","JCRName":"MATHEMATICS","Score":null,"Total":0}
On power integral bases of certain pure number fields defined by $x^{84}-m$
Let $K$ be a pure number field generated by a complex root of a monic irreducible polynomial $F(x)=x^{60}-m\in \mathbb{Z}[x]$, with $m\neq \pm1$ a square free integer. In this paper, we study the monogeneity of $K$. We prove that if $m\not\equiv 1\md{4}$, $m\not\equiv \mp 1 \md{9} $ and $\overline{m}\not\in\{\mp 1,\mp 7\} \md{25}$, then $K$ is monogenic. But if $m\equiv 1\md{4}$, $m\equiv \mp1 \md{9}$, or $m\equiv \mp 1\md{25}$, then $K$ is not monogenic. Our results are illustrated by examples.
期刊介绍:
Revista de la Unión Matemática Argentina is an open access journal, free of charge for both authors and readers. We publish original research articles in all areas of pure and applied mathematics.