Irreducible skew polynomials over domains

TL;DR

This paper establishes criteria for the irreducibility of low-degree skew polynomials over domains, with applications to quantized Weyl algebras and quantum planes, advancing understanding of their algebraic structure.

Contribution

It provides new criteria for irreducibility of skew polynomials of degree up to four over domains, including specific applications to quantum algebra structures.

Findings

Criteria for irreducibility of degree ≤ 4 skew polynomials

Application to quantized Weyl algebras

Analysis of polynomials of the form t^m - a

Abstract

Let $S$ be a domain and $R=S[t;\sigma,\delta]$ a skew polynomial ring, where $\sigma$ is an injective endomorphism of $S$ and $\delta$ a left $\sigma$ -derivation. We give criteria for skew polynomials $f\in R$ of degree less or equal to four to be irreducible. We apply them to low degree polynomials in quantized Weyl algebras and the quantum planes. We also consider $f(t)=t^m-a\in R$.