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 $f$.