A BRIEF NOTE ON SOME INFINITE FAMILIES OF MONOGENIC POLYNOMIALS. Issue 2 (13th February 2019)

• Record Type: Journal Article
• Authors: JONES, LENNY
• Is Part Of: Bulletin of the Australian Mathematical Society. Volume 100:Issue 2(2019)
• Page Start: 239
• Page End: 244
• Publication Date: 2019-02-13
• Subjects: primary 11R04, -- secondary 11R09, -- 12F05; discriminant, -- monogenic, -- irreducible; Mathematics -- Societies, etc; Mathematics -- Periodicals; 510.5
• DOI: 10.1017/S0004972719000182
• Languages: English
• ISSNs: 0004-9727

Abstract

Abstract : 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 $f(\unicode[STIX]{x1D703})=0$ .