On the discriminant and index of a certain class of polynomials

Abstract

Let $f(x)=(x^2+1{)}^n-a{x}^n\in \mathbb{Z}[x]$ and assume $f(x)$ is irreducible. Let $\theta$ be a root of $f(x)$, set $K=\mathbb{Q}(\theta )$, and denote by ${\mathbb{Z}}_K$ the ring of integers of $K$. The index of $f$, denoted $\mathrm{ind}(f)$, is the index of $\mathbb{Z}[\theta ]$ in ${\mathbb{Z}}_K$. A polynomial $f(x)$ is said to be monogenic if $\mathrm{ind}(f)=1$. In this article, we explicitly compute the discriminant of the polynomial $f(x)$, and then derive necessary and sufficient conditions on the parameters $a$ and $n$ for $f(x)$ to be monogenic. Furthermore, we provide a complete description of the primes that divide $\mathrm{ind}(f)$.

To appear in Bulletin Australian Math. Soc