The norm of a skew polynomial

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 $f$.