Monogenic cyclotomic compositions

TL;DR

This paper proves the irreducibility of a specific cyclotomic polynomial composition and establishes a basis for the ring of integers in the corresponding number field, advancing understanding of cyclotomic polynomial structures.

Contribution

It demonstrates the irreducibility of monogenic cyclotomic compositions and explicitly constructs a basis for their rings of integers.

Findings

$T(x)$ is irreducible over $\\mathbb{Q}$.

The set $\\{1,\ heta,\ heta^2,\\ldots,\ heta^{2^{n-1}p^{m-1}(p-1)-1}\\ ight\\}$ forms a basis for the ring of integers.

The results apply to compositions of cyclotomic polynomials of prime power and power of two.

Abstract

Let $m$ and $n$ be positive integers, and let $p$ be a prime. Let $T(x)=\Phi_{p^m}\left(\Phi_{2^n}(x)\right)$, where $\Phi_k(x)$ is the cyclotomic polynomial of index $k$. In this article, we prove that $T(x)$ is irreducible over $\mathbb Q$ and that \[\left\{1,\theta,\theta^2,\ldots,\theta^{2^{n-1}p^{m-1}(p-1)-1}\right\}\] is a basis for the ring of integers of $\mathbb Q(\theta)$, where $T(\theta)=0$.