{"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 $f(x)\in {\mathbb Z}[x]$ of degree N is called monogenic if $f(x)$ is irreducible over ${\mathbb Q}$ and $\{1,\theta ,\theta ^2,\ldots ,\theta ^{N-1}\}$ is a basis for the ring of integers of ${\mathbb Q}(\theta )$, where $f(\theta )=0$. We prove that there exist exactly three distinct monogenic trinomials of the form $x^4+bx^2+d$ 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.
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.
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Published for the Australian Mathematical Society