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Infinite families of monogenic quadrinomials, quintinomials and sextinomials

Volume 169 / 2022

Lenny Jones Colloquium Mathematicum 169 (2022), 1-10 MSC: Primary 11R04; Secondary 12F05. DOI: 10.4064/cm8552-4-2021 Published online: 21 January 2022


Let $f(x)\in \mathbb Z [x]$ be monic, with $\deg (f)=n$. We say $f(x)$ is \emph {monogenic} if $f(x)$ is irreducible over $\mathbb Q $ and $\{1,\alpha ,\alpha ^2,\ldots , \alpha ^{n-1}\}$ is a basis for the ring of integers of $K=\mathbb Q (\alpha )$, where $f(\alpha )=0$. In this article, we derive a new polynomial discriminant formula, and we use it to construct infinite families of monogenic quadrinomials, quintinomials and sextinomials for any degree $n\ge 3,4,5$, respectively. These results extend previous work of the author. We also give a brief discussion concerning the adaptation of our approach beyond sextinomials.


  • Lenny JonesProfessor Emeritus
    Department of Mathematics
    Shippensburg University
    1871 Old Main Drive
    Shippensburg, PA 17257, USA

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