The norm of a skew polynomial

TL;DR

This paper studies the properties of the norm function in skew polynomial rings over division algebras, focusing on how the norm relates to polynomial reducibility and the structure of the central quotient algebra.

Contribution

It provides explicit calculations of the norm for skew polynomials and explores the connection between polynomial reducibility and the reducibility of their norms.

Findings

Calculated the norm for specific skew polynomials.

Established conditions linking polynomial reducibility to norm reducibility.

Analyzed the structure of the central quotient algebra.

Abstract

Let $D$ be a finite-dimensional division algebra over its center and $R=D[t;\sigma,\delta]$ a skew polynomial ring. Under certain assumptions on $\delta$ and $\sigma$, the ring of central quotients $D(t;\sigma,\delta) = \{f/g \,|\, f \in D[t;\sigma,\delta], g \in C(D[t;\sigma,\delta])\}$ of $D[t;\sigma,\delta]$ is a central simple algebra with reduced norm $N$. We calculate the norm $N(f)$ for some skew polynomials $f\in R$ and investigate when and how the reducibility of $N(f)$ reflects the reducibility of $f$.