= Coefficient pairing for a monogenic algebra
{title2=$\lambda(ab)=[X^{n-1}](ab\bmod g)$}
For a monic degree-$n$ <polynomial> $g$ over a <commutative ring> $R$, let $A=R[X]/(g)$ and let $\lambda$ take the coefficient of $X^{n-1}$ in the reduced representative. The pairing $(a,b)\mapsto\lambda(ab)$ is unimodular on $1,X,\ldots,X^{n-1}$: reversing one basis order gives a triangular matrix with diagonal ones. When $A$ embeds in a separable field extension, <Lagrange interpolation> identifies $\lambda(a)$ with $\operatorname{Tr}(a/g'(x))$. This yields the derivative formula for the <inverse different>.
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