MONOGENIC EVEN QUARTIC TRINOMIALS

IF 0.6 4区 数学 Q3 MATHEMATICS
LENNY JONES
{"title":"MONOGENIC EVEN QUARTIC TRINOMIALS","authors":"LENNY JONES","doi":"10.1017/s0004972724000510","DOIUrl":null,"url":null,"abstract":"<p>A monic polynomial <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240912125559562-0272:S0004972724000510:S0004972724000510_inline1.png\"><span data-mathjax-type=\"texmath\"><span>$f(x)\\in {\\mathbb Z}[x]$</span></span></img></span></span> of degree <span>N</span> is called <span>monogenic</span> if <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240912125559562-0272:S0004972724000510:S0004972724000510_inline2.png\"><span data-mathjax-type=\"texmath\"><span>$f(x)$</span></span></img></span></span> is irreducible over <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240912125559562-0272:S0004972724000510:S0004972724000510_inline3.png\"><span data-mathjax-type=\"texmath\"><span>${\\mathbb Q}$</span></span></img></span></span> and <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240912125559562-0272:S0004972724000510:S0004972724000510_inline4.png\"><span data-mathjax-type=\"texmath\"><span>$\\{1,\\theta ,\\theta ^2,\\ldots ,\\theta ^{N-1}\\}$</span></span></img></span></span> is a basis for the ring of integers of <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240912125559562-0272:S0004972724000510:S0004972724000510_inline5.png\"><span data-mathjax-type=\"texmath\"><span>${\\mathbb Q}(\\theta )$</span></span></img></span></span>, where <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240912125559562-0272:S0004972724000510:S0004972724000510_inline6.png\"><span data-mathjax-type=\"texmath\"><span>$f(\\theta )=0$</span></span></img></span></span>. We prove that there exist exactly three distinct monogenic trinomials of the form <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240912125559562-0272:S0004972724000510:S0004972724000510_inline7.png\"><span data-mathjax-type=\"texmath\"><span>$x^4+bx^2+d$</span></span></img></span></span> whose Galois group is the cyclic group of order 4. We also show that the situation is quite different when the Galois group is not cyclic.</p>","PeriodicalId":50720,"journal":{"name":"Bulletin of the Australian Mathematical Society","volume":"65 1","pages":""},"PeriodicalIF":0.6000,"publicationDate":"2024-09-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Bulletin of the Australian Mathematical Society","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1017/s0004972724000510","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: 0

Abstract

A monic polynomial Abstract Image$f(x)\in {\mathbb Z}[x]$ of degree N is called monogenic if Abstract Image$f(x)$ is irreducible over Abstract Image${\mathbb Q}$ and Abstract Image$\{1,\theta ,\theta ^2,\ldots ,\theta ^{N-1}\}$ is a basis for the ring of integers of Abstract Image${\mathbb Q}(\theta )$, where Abstract Image$f(\theta )=0$. We prove that there exist exactly three distinct monogenic trinomials of the form Abstract Image$x^4+bx^2+d$ whose Galois group is the cyclic group of order 4. We also show that the situation is quite different when the Galois group is not cyclic.

单向偶四次方三项式
如果 $f(x)$ 在 ${\mathbb Q}$ 上是不可约的,并且 ${1、\theta ,\theta ^2,\ldots ,\theta ^{N-1}\}$ 是 ${\mathbb Q}(\theta )$ 的整数环的基,其中 $f(\theta )=0$.我们证明恰好存在三个不同的形式为 $x^4+bx^2+d$ 的单元三项式,它们的伽罗瓦群是阶数为 4 的循环群。我们还证明了当伽罗瓦群不是循环群时,情况会截然不同。
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来源期刊
CiteScore
1.20
自引率
14.30%
发文量
149
审稿时长
4-8 weeks
期刊介绍: Bulletin of the Australian Mathematical Society aims at quick publication of original research in all branches of mathematics. Papers are accepted only after peer review but editorial decisions on acceptance or otherwise are taken quickly, normally within a month of receipt of the paper. The Bulletin concentrates on presenting new and interesting results in a clear and attractive way. Published Bi-monthly Published for the Australian Mathematical Society
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