On the index of power compositional polynomials

TL;DR

The paper investigates monogenicity of power-compositional polynomials $f(x^k)$ where $f$ is monic irreducible in $ obreak \mathbb{Z}[x]$. It proves a concise necessary-and-sufficient criterion: if $f(x^k)$ is irreducible, then $f(x^k)$ is monogenic precisely when $f(x)$ is monogenic, no prime $p|k$ divides the index $Ind(f(x^p))$, and $f(0)$ is squarefree. This criterion shows that monogenity of $f(x^k)$ does not depend on the prime-power structure of $k$ and yields infinite families of monogenic polynomials, with concrete results for polynomials of the form $f(x)=x^d+A\cdot h(x)$. The paper develops a robust framework based on Dedekind's criterion, Uchida's reformulation, and discriminant relations to control which primes can contribute to the index and to construct explicit infinite families of monogenic polynomials.

Abstract

The index of a monic irreducible polynomial $f(x)\in\mathbb{Z}[x]$ having a root $θ$ is the index $[\mathbb{Z}_K:\mathbb{Z}[θ]]$, where $\mathbb{Z}_K$ is the ring of algebraic integers of the number field $K=\mathbb{Q}(θ)$. If $[\mathbb{Z}_K:\mathbb{Z}[θ]]=1$, then $f(x)$ is monogenic. In this paper, we give necessary and sufficient conditions for a monic irreducible power compositional polynomial $f(x^k)$ belonging to $\mathbb{Z}[x]$, to be monogenic. As an application of our results, for a polynomial $f(x)=x^d+A\cdot h(x)\in\mathbb{Z}[x],$ with $d>1, \operatorname{deg} h(x)<d$ and $|h(0)|=1$, we prove that for each positive integer $k$ with $\operatorname{rad}(k)\mid \operatorname{rad}(A)$, the power compositional polynomial $f(x^k)$ is monogenic if and only if $f(x)$ is monogenic, provided that $f(x^k)$ is irreducible. At the end of the paper, we give infinite families of polynomials as examples.

Figures (1)

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Theorems & Definitions (45)

• Theorem 1.1
• Remark 1.2
• Remark 1.3
• Corollary 1.4
• Corollary 1.5
• Theorem 1.6
• Remark 1.7
• Corollary 1.8
• Theorem 1.9
• Lemma 2.1