Corollary 2

— Suppose that $A$ is a field and let $f, g \in A[X]$, then the following conditions are equivalent when $f, g$ are non-zero: (i) $\operatorname{res}(f, g) \neq 0$; (ii) the polynomials $f$ and $g$ are relatively prime in $A[X]$; \* (iii) for any extension L of A the polynomials $f$ and $g$ have no common root in L. \*

We can reduce at once to the case where $f$ is monic (IV, p. 77, remark 3). The equivalence of (i) and (ii) is just a translation of Cor. 1, by Def. 1 of IV, p. 12; the equivalence of (ii) and (iii) is just Cor. 7 of IV, p. 13.