The discriminant of a given polynomial is a number, calculated from the coefficients of that polynomial, that vanishes if and only if that polynomial has one or more multiple roots. Using the discriminant we can test for the presence of multiple roots, without having to actually calculate the roots of the polynomial in question.

There are other ways to do this of course; one can look at the formal derivative of the polynomial (it will be coprime to the original polynomial if and only if that original had no multiple roots). But the discriminant turns out to be valuable in a number of other contexts. For example, we will see that the discriminant of $X^{2}+bX+c$ is $b^{2}-4c$; the quadratic formula states that the roots are $-b/2\pm\sqrt{b^{2}-4c}/2$, so that the discriminant also determines whether the roots of this polynomial are real or not. In higher degrees, its role is more complicated.

There are other uses of the word “discriminant” that are closely related to this one. If $\mathbb{Q}(\alpha)$ is a number field, then the discriminant of $\mathbb{Q}(\alpha)$ is the discriminant of the minimal polynomial of $\alpha$. For more general extensions of number fields, one must use a different definition of discriminant generalizing this one. If we have an elliptic curve over the rational numbers defined by the equation $y^{2}=x^{3}+Ax+B$, then its modular discriminant is the discriminant of the cubic polynomial on the right-hand side. For more on both these facts, see [1] on number fields and [2] on elliptic curves.

The discriminant of order $n\in\mathbb{N}$ is the polynomial, denoted here ¹¹ The discriminant of a polynomial $p$ is oftentimes also denoted as “$\mathop{\rm disc}(p)$” by $\delta^{{(n)}}=\delta^{{(n)}}(a_{1},\ldots,a_{n})$, characterized by the following relation:

where

is the $k^{{\text{th}}}$ elementary symmetric polynomial.

The above relation is a defining one, because the right-hand side of (1) is, evidently, a symmetric polynomial, and because the algebra of symmetric polynomials is freely generated by the basic symmetric polynomials, i.e. every symmetric polynomial arises in a unique fashion as a polynomial of $s^{1},\ldots,s^{n}$.

Let $\mathbb{K}$ be a field, let $x$ denote an indeterminate, and let

be a monic polynomial over $\mathbb{K}$. We define $\delta[p]$, the discriminant of $p$, by setting

The discriminant of a non-monic polynomial is defined homogenizing the above definition, i.e by setting

Here are the first few discriminants.

Here is the matrix used to calculate $\delta^{{(4)}}$: