On the discriminant and index of a certain class of polynomials

TL;DR

This paper investigates the algebraic properties of a specific class of polynomials, explicitly computes their discriminants, and determines conditions under which these polynomials are monogenic, including prime divisors of their index.

Contribution

It provides explicit discriminant formulas and necessary and sufficient conditions for monogenicity of the polynomial class, along with a prime divisor analysis of the index.

Findings

Explicit discriminant formulas for the polynomial class.

Necessary and sufficient conditions for monogenicity.

Complete characterization of primes dividing the index.

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 $\operatorname{ind}(f)$, is the index of $\mathbb{Z}[\theta]$ 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 $\operatorname{ind}(f)$.