discriminant

《A Course in Finite Group Representation Theory》 · 第 166—167 页 · Section 4.4「Symmetric Polynomials, Discriminant, Resultant」

Working over the coefficient domain $R=\mathbb Z$, consider $\prod_{i<j}(T_i-T_j)^2$ as a symmetric polynomial in $\mathbb Z[T_1,\ldots,T_n]$. The latter is a polynomial $\Delta$ in the elementary symmetric polynomials $s_1,\ldots,s_n$, as we can read from Theorem 1. One calls $\Delta=\Delta(s_1,\ldots,s_n)$ the discriminant of the polynomial $\prod_{i=1}^n(X-T_i)=\sum_{j=0}^n(-1)^js_jX^{n-j}$, where $X$ is viewed as a variable over the ring $\mathbb Z[T_1,\ldots,T_n]$. Now, to define the discriminant of any monic polynomial $f=X^n+c_1X^{n-1}+\cdots+c_n\in R[X]$ over any coefficient ring $R$, set $\Delta_f=\Delta(-c_1,c_2,\ldots,(-1)^nc_n)$.