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 $θ$ be a root of $f(x)$, set $K= \mathbb{Q}(θ)$, and denote by $\mathbb{Z}_{K}$ the ring of integers of $K$. The index of $f$, denoted $\operatorname{ind}(f)$, is the index of $\mathbb{Z}[θ]$ in $\mathbb{Z}_{K}$. A polynomial $f(x)$ is said to be monogenic if $\operatorname{ind}(f) = 1$. 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 $\operatorname{ind}(f)$.

Theorems & Definitions (10)

• Theorem 1.1
• Theorem 1.2
• Corollary 1.3
• Theorem 2.1
• Proposition 2.2
• proof : Proof of Theorem \ref{['thm-1.1']}
• proof : Proof of Theorem \ref{['Thm-1.2']}
• Example 3.1
• Remark 3.2
• Remark 3.3