Generating Infinite Families of Monogenic Polynomials Using a New Discriminant Formula

Recently, Otake and Shaska have given a formula for the discriminant of quadrinomials of the form f(x) = xⁿ + t(x² + ax + b). In their concluding remarks, they ask if a formula can be found for the discriminant of f(x) = xⁿ + tg(x) when n > (g) = 3. Assuming that f(x) = xⁿ + tg(x) is irreducible, and under certain restrictions on a polynomial related to $g(x)$, in this article we give a formula for the discriminant of $f(x)$, regardless of (g) ≥ 1. We then use our discriminant formula to generate some new infinite families of monogenic polynomials f(x) = xⁿ + tg(x) with n > (g), when $g(x)$ is monic and (g)∈ 2 \2, 3, 4\.