Discriminant of a polynomial

Definition

Let $K$ be a field and let $f = a_n x^n + \dots + a_0 \in K[x]$ with $n \ge 1$ and $a_n \neq 0$. Let $f = a_n (x - r_1) \cdots (x - r_n)$ in a splitting field of $f$. The discriminant of $f$ is $$\operatorname{Disc}(f) = a_n^{2n-2} \prod_{1 \le i < j \le n} (r_i - r_j)^2.$$

Remarks

The product is symmetric in the roots, so $\operatorname{Disc}(f) \in K$, by the fundamental theorem of symmetric polynomials. It is a polynomial with integer coefficients in $a_0, \dots, a_n$. So $\operatorname{Disc}(f) = 0$ if and only if $f$ has a repeated root. For $f = a x^2 + b x + c$, $\operatorname{Disc}(f) = b^2 - 4ac$. For $f = x^3 + p x + q$, $\operatorname{Disc}(f) = -4p^3 - 27q^2$.

Explanations

L1 human

The discriminant measures how far apart the roots of a polynomial are. It is zero when two roots coincide. For a real quadratic, its sign tells whether the roots are real or complex.