Proposition 9

— Let $p_1$ and $q_1$ be positive integers and $f_1, g_1$ polynomials in $A[X]$ such that $\deg f_1 \leq p_1, \deg g_1 \leq q_1$; then we have

We have $\text{res}_{p, q}(f, g) = (-1)^{pq} \text{res}_{q, p}(g, f)$ (IV, p. 76); so it is enough to prove (35). Likewise, Remark 3 (loc. cit.) shows that if formula (35) is established for a polynomial $f$, then it holds for all polynomials of the form Af, where $A \in A$. Finally, when f is monic of degree p, formula (35) follows from Prop. 7 (IV, p. 77) by the formula $N_{E/A}(ab) = N_{E/A}(a) \cdot N_{E/A}(b)$. In all we conclude that (35) holds when the coefficient $t_p$ of $X^p$ in f is invertible.