Discriminant of a polynomial

Let \[ f(x)=a_n\prod_{i=1}^n(x-r_i) \] in a splitting field, with \(a_n\ne0\). The discriminant of \(f\) is \[ \operatorname{Disc}(f) = a_n^{2n-2}\prod_{1\le i<j\le n}(r_i-r_j)^2. \]

Equivalently, \[ \operatorname{Disc}(f) = (-1)^{n(n-1)/2}a_n^{-1}\operatorname{Res}(f,f'). \] Because the product of squared root differences is symmetric in the roots, the fundamental theorem of symmetric polynomials shows that the discriminant is a polynomial in the coefficients of \(f\).

Over a field, one has \[ \operatorname{Disc}(f)=0 \iff f\text{ has a repeated root} \iff \gcd(f,f')\ne1. \] For a monic quadratic \(x^2+bx+c\), this gives \(\operatorname{Disc}(f)=b^2-4c\).