Corollary 1

— Let $f \in A[X]$ be a monic polynomial; for every polynomial $g \in A[X]$ the following conditions are equivalent: (i) $\operatorname{res}(f, g)$ is invertible in $A$; (ii) there exist polynomials $u, v$ in $A[X]$ such that $uf + vg = 1$; (iii) $g(x)$ is invertible in the algebra $A[X]/(f)$. The equivalence of (i) and (iii) follows from Prop. 3 of III, p. 545 and Prop. 7; that of (ii) and (iii) follows trivially.