---
tag: 0CCPN09EMJ
kind: definition
name: Discriminant of a polynomial
msc: ["12E05"]
defines: 0VVK64Q4JK
uses: [0SHMC00RJV, 0GW1ZZH0R4, 067F8YVYMH]
sources:
  - url: https://en.wikipedia.org/wiki/Discriminant
    locator: Expression in terms of the roots
  - url: https://en.wikipedia.org/wiki/Discriminant
    locator: Low degrees
---

## Definition

Let $K$ be a [[field|0QA8CHH8MG]] 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|02NZZ1J0FH]] 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|0GW1ZZH0R4]]. 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|067F8YVYMH]].
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.