A BRIEF NOTE ON SOME INFINITE FAMILIES OF MONOGENIC POLYNOMIALS. Issue 2 (13th February 2019) - JONES, LENNY
Abstract : Suppose that $f(x)=x^{n}+A(Bx+C)^{m}\in \mathbb{Z}[x]$, with $n≥ 3$ and $1≤ m<n$, 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 $\{1, \unicode[STIX]{x1D703}, \unicode[STIX]{x1D703}^{2}, …, \unicode[STIX]{x1D703}^{n-1}\}$ is an integral basis for the ring of integers of $\mathbb{Q}(\unicode[STIX]{x1D703})$, where $f(\unicode[STIX]{x1D703})=0$ .