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A BRIEF NOTE ON SOME INFINITE FAMILIES OF MONOGENIC POLYNOMIALS

Part of: Field extensions Algebraic number theory: global fields

Published online by Cambridge University Press:  13 February 2019

LENNY JONES Open the ORCID record for LENNY JONES [Opens in a new window]
LENNY JONES*
Affiliation:
Department of Mathematics, Shippensburg University, Shippensburg, PA 17257, USA email lkjone@ship.edu
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Abstract

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Suppose that $f(x)=x^{n}+A(Bx+C)^{m}\in \mathbb{Z}[x]$, with $n\geq 3$ and $1\leq m<n$, is irreducible over $\mathbb{Q}$. By explicitly calculating the discriminant of $f(x)$, we prove that, when $\gcd (n,mB)=C=1$, there exist infinitely many values of $A$ such that the set $\{1,\unicode[STIX]{x1D703},\unicode[STIX]{x1D703}^{2},\ldots ,\unicode[STIX]{x1D703}^{n-1}\}$ is an integral basis for the ring of integers of $\mathbb{Q}(\unicode[STIX]{x1D703})$, where $f(\unicode[STIX]{x1D703})=0$.

Keywords

discriminantmonogenicirreducible

MSC classification

Primary: 11R04: Algebraic numbers; rings of algebraic integers
Secondary: 11R09: Polynomials (irreducibility, etc.) 12F05: Algebraic extensions
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 100 , Issue 2 , October 2019 , pp. 239 - 244
Copyright
© 2019 Australian Mathematical Publishing Association Inc. 

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