A BRIEF NOTE ON SOME INFINITE FAMILIES OF MONOGENIC POLYNOMIALS

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