Generating infinite families of monogenic polynomials using a new discriminant formula. (English) Zbl 1441.11268
Summary: Recently, S. Otake and T. Shaska [Contemp. Math. 724, 55–72 (2019; Zbl 1423.12011)] have given a formula for the discriminant of quadrinomials of the form \(f(x) =x^n+t(x^2+ax+b)\). In their concluding remarks, they ask if a formula can be found for the discriminant of \(f(x) =x^n+tg(x)\) when \(n >\mathrm{deg}(g) = 3\). Assuming that \(f(x) =x^n+tg(x)\) 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 \(\mathrm{deg}(g)\geq 1\). We then use our discriminant formula to generate some new infinite families of monogenic polynomials \(f(x) =x^n+tg(x)\) with \(n >\mathrm{deg}(g)\), when \(g(x)\) is monic and \(\mathrm{deg}(g)\in \{2,3,4\}\).
MSC:
| 11R04 | Algebraic numbers; rings of algebraic integers |
| 11R09 | Polynomials (irreducibility, etc.) |
| 12F05 | Algebraic field extensions |
Citations:
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