Proposition 10

Let $m \geq 1$. There exists a unique polynomial $A \in \mathbf{Z}[A_1, \ldots, A_m]$ with the following property : for any commutative ring A and monic polynomial $f = X^m + \sum_{i=1}^m a_i X^{m-i}$ in $A[X]$ we have

Moreover A is of degree $\leq 2m-2$ and if we assign weight i to $A_i$ then A is isobaric of weight $m(m-1)$. a) Uniqueness of A : if A satisfies (48), we have in particular $A = \operatorname{dis}(F)$, where F is the polynomial $X^m + \sum_{i=1}^m A_i X^{m-i}$ with coefficients in $\mathbf{Z}[A_1, \ldots, A_m]$. b) Existence of A : let $s_1, \ldots, s_m$ be the elementary symmetric polynomials in the indeterminates $X_1, \ldots, X_m$. There exists a polynomial $A \in \mathbf{Z}[A_1, \ldots, A_m]$, isobaric of weight $m(m-1)$, such that

for the right-hand side is a symmetric polynomial P, homogeneous of degree m(m − 1) in $Z[X_1, ..., X_m]$ (IV, p. 62, Th. 1). Now formula (46) means that dis(f) = P (f ), in the notation of IV, p. 74, and (48) follows directly from this.

c) Degree of A : the relation (47) and the definition of the resultant (IV, p. 76) imply the formula

with the following values of the $a_{ij}$

$a_{0j} = 0$ if $j \neq 0 , \ j \neq m-1$

In these formulae it is understood that $A_i = 1$ and $A_i = 0$ for $i < 0$ or $i > m$. Now (50) shows at once that A is of degree $\leq 2m-2$, as we had to prove.

Prop. 10 allows us to extend the definition of the discriminant to non-monic polynomials. Let $m \geq 1$ be an integer, then there exists a unique homogeneous polynomial of degree $2m-2$, $\tilde{\Delta}$ say, in $Z[A_0, A_1, ..., A_m]$ such that

for, since A is of degree $\leq 2m-2$, the rational fraction

belongs to the subring $Z[A_0, A_1, ..., A_m]$ of $Q(A_0, A_1, ..., A_m)$. If $A_i$ has weight i for $0 \leq i \leq m$, then $\tilde{\Delta}$ is isobaric of weight $m(m-1)$. Iff is a polynomial of degree $\leq m$, say

we shall put

When $m = \deg f$, we shall simply write dis(f) for $\operatorname{dis}_m(f)$; if f is monic, dis(f) agrees with the discriminant defined in Def. 2, by (48), (51) and (52).