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 2018-05-22, 07:17 #2 Nick     Dec 2012 The Netherlands 110110111112 Posts 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. Last fiddled with by Nick on 2018-05-22 at 07:18 Reason: Typo