Definition 2

Let $f$ be a monic polynomial of $A[X]$ of degree $m$, and denote by $E$ the $A$-algebra $A[X]/(f)$ and by $x$ the canonical image of $X$ in $E$. We define the discriminant of $f$, written $\operatorname{dis}(f)$, as the discriminant $D_{E/A}(1, x, \ldots, x^{m-1})$ of the basis $(1, x, \ldots, x^{m-1})$ of the $A$-algebra $E$.

For every positive integer $k$ we write $\rho_k = \operatorname{Tr}_{E/A}(x^k)$. By III, p. 549, Def. 2 amounts to the formula

*Examples. — 1) If $f$ is a monic polynomial of degree 0 or 1, we have $\operatorname{dis}(f) = 1$, by (45). 2) Let $f(X) = X^2 + \alpha X + \beta$ be a monic polynomial of degree 2. Newton's relations may be written in the form (IV, p. 75)

whence $p_1 = -a$, $p_2 = a^2 - 2p$. It follows that we have

Let B be a commutative ring, $\rho$ a homomorphism from A to B and $\xi_1, \ldots, \xi_m$ elements of B such that

Let us write M for the matrix $(\rho(p_{i + j}))_{0 \leq i, j < m}$ and D for the Van der Monde matrix $(\xi_i^{j+1})_{0 \leq i, j < m}$. By Example 4 of IV, p. 75 we have whence $M = 'D . D ;$ by III, p. 532 we have $D = \prod_{i > j} (\xi_i - \xi_j)$ and $\det M = (\det D)^2$, that is Moreover (on writing D for the derivation $\frac{d}{dX}$) we have for $1 \leq i \leq m$, whence By Cor. 1 of IV, p. 80 applied to the universal decomposition of $f$, we finally obtain