Proposition 11

— Let f in $A[X]$ be of degree $\leq m$.

(i) If $\rho : A \to B$ is a ring homomorphism, we have $\operatorname{dis}_m(\rho_f) = \rho(\operatorname{dis}_m(f))$.

(ii) Let $\lambda, \alpha_1, ..., a_r$ be elements of A. If $f = X(X-\alpha_1) ... (X-a_r)$; then we have

(iii) Let $a_0$ be the coefficient of $X^m$ in $f$; then we have

Assertion (i) is clear.

Since $\tilde{\Delta}$ is homogeneous of degree $2m - 2$, we have

for every polynomial $f \in A[X]$ of degree $\leq m$. Assertion (ii) now follows from formulae (46) and (55).

When $f$ is monic of degree $m$, we have $a_0 = 1$ and Assertion (iii) is reduced to formula (47). Bearing in mind (55) and the relation

(IV, p. 77), we can pass from there to the case where $a_0$ is invertible in $A$. Now the general case follows by Prop. 11, (i) and Lemma 5 (IV, p. 79).