The discriminantMathworldPlanetmathPlanetmathPlanetmathPlanetmath of a given polynomialMathworldPlanetmathPlanetmathPlanetmath 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 coprimeMathworldPlanetmathPlanetmath 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 X2+bX+c is b2-4c; the quadratic formula states that the roots are -b/2±b2-4c/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 (α) is a number fieldMathworldPlanetmath, then the http://planetmath.org/node/2895discriminant of (α) is the discriminant of the minimal polynomialPlanetmathPlanetmath of α. For more general extensions of number fields, one must use a different definition of discriminant generalizing this one. If we have an elliptic curveMathworldPlanetmath over the rational numbers defined by the equation y2=x3+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 is the polynomial, denoted here 11 The discriminant of a polynomial p is oftentimes also denoted as “disc(p) by δ(n)=δ(n)(a1,,an), characterized by the following relation:

δ(n)(s1,s2,,sn)=i=1nj=i+1n(xi-xj)2, (1)



is the kth elementary symmetric polynomial.

The above relation is a defining one, because the right-hand side of (1) is, evidently, a symmetric polynomialMathworldPlanetmath, and because the algebraMathworldPlanetmathPlanetmath 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 s1,,sn.

Proposition 1.

The discriminant d of a polynomial may be expressed with the resultantMathworldPlanetmath R of the polynomial and its first derivativeMathworldPlanetmath:


Proposition 2.

Up to sign, the discriminant is given by the determinantMathworldPlanetmath of a 2n-1 square matrixMathworldPlanetmath with columns 1 to n-1 formed by shifting the sequence1,a1,,an,  and columns n to 2n-1 formed by shifting the sequence  n,(n-1)a1,,an-1,  i.e.

δ(n)=|100n000a110(n-1)a1n00an-2an-312an-23an-3n0an-1an-2a1an-12an-2(n-1)a1nanan-1a20an-1(n-2)a2(n-1)a100an-100an-12an-200an000an-1| (2)

Multiple root test.

Let 𝕂 be a field, let x denote an indeterminate, and let


be a monic polynomialMathworldPlanetmath over 𝕂. We define δ[p], the discriminant of p, by setting


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

Proposition 3.

The discriminant vanishes if and only if p has multiple roots in its splitting fieldMathworldPlanetmath.


It isn’t hard to show that a polynomial has multiple roots if and only if that polynomial and its derivative share a common root. The desired conclusion now follows by observing that the determinant formula in equation (2) gives the resolvent of a polynomial and its derivative. This resolvent vanishes if and only if the polynomial in question has a multiple root. ∎

Some Examples.

Here are the first few discriminants.

δ(1) =1
δ(2) =a12-4a2
δ(3) =18a1a2a3+a12a22-4a23-4a13a3-27a32
δ(4) =a12a22a32-4a23a32-4a13a33+18a1a2a33-27a34

Here is the matrix used to calculate δ(4):



  • 1 Daniel A. Marcus, Number Fields, Springer, New York.
  • 2 Joseph H. Silverman, The Arithmetic of Elliptic Curves. Springer-Verlag, New York, 1986.

See also the bibliography for number theory (http://planetmath.org/BibliographyForNumberTheory) and the bibliography for algebraic geometry (http://planetmath.org/BibliographyForAlgebraicGeometry).

Title discriminant
Canonical name Discriminant
Date of creation 2013-03-22 12:31:12
Last modified on 2013-03-22 12:31:12
Owner rspuzio (6075)
Last modified by rspuzio (6075)
Numerical id 17
Author rspuzio (6075)
Entry type Definition
Classification msc 12E05
Synonym polynomial discriminant
Related topic Resolvent
Related topic DiscriminantOfANumberField
Related topic ModularDiscriminant
Related topic JInvariant
Related topic DiscriminantOfAlgebraicNumber