The norm of a skew polynomial

Let $D$ be a finite-dimensional division algebra over its center and $R=D[t;σ,δ]$ a skew polynomial ring. Under certain assumptions on $δ$ and $σ$, the ring of central quotients $D(t;σ,δ) = \{f/g \,|\, f \in D[t;σ,δ], g \in C(D[t;σ,δ])\}$ of $D[t;σ,δ]$ 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 $f$.

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