[Nick]

Let R be a UFD with K its field of fractions and f in R[X] monic and irreducible of degree n≥1.
Let α be in an algebraic closure of K satisfying f(α)=0.
Then
\[ 1,\alpha,\alpha^2,\ldots,\alpha^{n-1} \]
is a basis for R[α] over R.
Now take any r,s in R with s≠0 and let g be the function from R[α] to R[α] given by g(x)=(r+sα)x.
By calculating the matrix for g relative to the above basis, we see that the norm (from R[α] to R) of r+sα is given by
\[ (-s)^nf\left(\frac{-r}{s}\right)\]
If f is not monic, then the first step fails already.