# Discriminant

A discriminant is a quantity computed from the coefficients of a polynomial, a quadratic form, or a ring extension that vanishes exactly when the object degenerates, in the polynomial case when two roots coincide. The name reflects the original purpose: it discriminates between equations whose solutions behave differently, without solving them.<sup>[1](https://mathshistory.st-andrews.ac.uk/Extras/Discriminant/)</sup>

| Key fact | Detail |
|---|---|
| Quadratic case | For \( ax^2+bx+c=0 \), \( \Delta = b^2-4ac \); two real roots if \( \Delta>0 \), one if \( \Delta=0 \), none if \( \Delta<0 \)<sup>[1](https://mathshistory.st-andrews.ac.uk/Extras/Discriminant/)</sup> |
| Cubic case | \( \Delta = b^2c^2-4ac^3-4b^3d-27a^2d^2+18abcd \); positive gives three distinct real roots, negative one real and two conjugate complex roots, zero a repeated root<sup>[1](https://mathshistory.st-andrews.ac.uk/Extras/Discriminant/)</sup> |
| General definition | The product of the squares of the differences of the polynomial's roots, equivalently (up to normalization) the resultant of the polynomial and its derivative<sup>[2](https://mathworld.wolfram.com/PolynomialDiscriminant.html)</sup><sup> • </sup><sup>[3](https://math.mit.edu/classes/18.721/notes/ag-dec1-2021.pdf)</sup> |
| Vanishing | The discriminant is zero if and only if the polynomial has a multiple root<sup>[3](https://math.mit.edu/classes/18.721/notes/ag-dec1-2021.pdf)</sup> |
| Term coined | James Joseph Sylvester, in a letter to Arthur Cayley dated 25 August 1851 and in print the same year<sup>[4](https://smf.emath.fr/system/files/filepdf/HUNGER-PARSHALL.pdf)</sup> |
| Number fields | The discriminant of a number field is the determinant of the trace-pairing matrix on an integral basis; a rational prime ramifies exactly when it divides it<sup>[5](https://websites.math.leidenuniv.nl/algebra/ant.pdf)</sup> |

## The quadratic and cubic cases

For a quadratic equation \( ax^2+bx+c=0 \), the discriminant \( \Delta = b^2-4ac \) is the expression under the square root in the [quadratic formula](https://www.edgepedia.org/quadratic-formula). Its sign classifies the real solutions: two distinct real solutions when \( \Delta>0 \), a single real solution when \( \Delta=0 \), and no real solution when \( \Delta<0 \). The expression has two summands, each of degree 2 in the coefficients.<sup>[1](https://mathshistory.st-andrews.ac.uk/Extras/Discriminant/)</sup>

For a cubic \( ax^3+bx^2+cx+d=0 \), the discriminant is

\[ \Delta = b^2c^2 - 4ac^3 - 4b^3d - 27a^2d^2 + 18abcd, \]

a sum of five terms each of degree 4. If \( \Delta>0 \) the equation has three distinct real solutions; if \( \Delta<0 \) it has one real and two conjugate complex solutions; if \( \Delta=0 \) all solutions are real but at least two coincide.<sup>[1](https://mathshistory.st-andrews.ac.uk/Extras/Discriminant/)</sup> For the depressed monic cubic \( x^3+ax^2+bx+c \), Sylvester's formula reduces to \( D(f) = -4a^3c + a^2b^2 + 18abc - 4b^3 - 27c^2 \), and after removing the quadratic term entirely, to the familiar \( 4b^3+27c^2 \) form.<sup>[6](https://eprints.whiterose.ac.uk/id/eprint/209753/1/Matrix%20factorizations%20of%20the%20discriminant%20of%20Sn.pdf)</sup>

## The general polynomial discriminant

For a polynomial of degree \( n \) with roots \( r_1, \dots, r_n \), the discriminant is the product of the squares of all root differences,

\[ \Delta = a_n^{\,2n-2} \prod_{i<j} (r_i - r_j)^2, \]

defined up to a constant factor, with several normalizations in use.<sup>[2](https://mathworld.wolfram.com/PolynomialDiscriminant.html)</sup> This product is a symmetric polynomial in the roots, so it can be rewritten as a polynomial in the coefficients alone. It vanishes precisely when two roots coincide, that is, when the polynomial has a multiple root; equivalently, the polynomial and its derivative share a common factor.<sup>[3](https://math.mit.edu/classes/18.721/notes/ag-dec1-2021.pdf)</sup> In terms of the resultant, the discriminant of \( p \) is the resultant of \( p \) and its derivative \( p' \), up to scaling by the leading coefficient and a sign \( (-1)^{n(n-1)/2} \).<sup>[2](https://mathworld.wolfram.com/PolynomialDiscriminant.html)</sup><sup> • </sup><sup>[7](https://arxiv.org/abs/1808.01264)</sup>

The number of monomial terms grows quickly with degree: 16 for degree 4, 59 for degree 5, and 246 for degree 6.<sup>[1](https://mathshistory.st-andrews.ac.uk/Extras/Discriminant/)</sup> Compact determinantal formulas therefore matter in practice. The terms can be obtained as determinants of Sylvester matrices, whose first \( n-1 \) rows hold the polynomial's coefficients and whose next \( n \) rows hold the coefficients of its first derivative.<sup>[1](https://mathshistory.st-andrews.ac.uk/Extras/Discriminant/)</sup> The most famous determinantal formula is due to Sylvester, with alternatives due to Bézout and Cayley; these matrices are equivalent in the sense of having isomorphic cokernels.<sup>[6](https://eprints.whiterose.ac.uk/id/eprint/209753/1/Matrix%20factorizations%20of%20the%20discriminant%20of%20Sn.pdf)</sup> [Computer algebra](https://www.edgepedia.org/computer-algebra) systems implement the discriminant directly; [Magma](https://www.edgepedia.org/magma-computer-algebra-system), for example, computes the discriminant of a multivariate polynomial viewed as a polynomial in one chosen variable.<sup>[8](https://magma.maths.usyd.edu.au/magma/handbook/text/249)</sup>

## History

**Sylvester's coinage.** In 1851, [James Joseph Sylvester](https://www.edgepedia.org/james-joseph-sylvester) (1814–1897), the English algebraist who also introduced the words matrix, invariant, and resultant, discovered a criterion for cubic equations that determines the number and type of solutions, and he named the expression involved the discriminant.<sup>[1](https://mathshistory.st-andrews.ac.uk/Extras/Discriminant/)</sup><sup> • </sup><sup>[9](https://mathshistory.st-andrews.ac.uk//Biographies/Sylvester/)</sup> He proposed the term in a letter to [Arthur Cayley](https://www.edgepedia.org/arthur-cayley) dated 25 August 1851, submitting it, along with invariant and resultant, for Cayley's approval and ratification, and used it in print for the first time that year in "On a Remarkable Discovery in the Theory of Canonical Forms and of Hyperdeterminants."<sup>[4](https://smf.emath.fr/system/files/filepdf/HUNGER-PARSHALL.pdf)</sup> Sylvester explained the choice: the discriminant "affords the discrimen or test" for ascertaining whether equal factors enter a function.<sup>[10](https://files.eric.ed.gov/fulltext/ED054941.pdf)</sup>

Before Sylvester, such expressions were called, after [Carl Friedrich Gauss](https://www.edgepedia.org/carl-friedrich-gauss), the "determinant of the function."<sup>[11](https://mathshistory.st-andrews.ac.uk//Miller/mathword/d/)</sup> The modern root-product definition was stated by [George Salmon](https://www.edgepedia.org/george-salmon) in the 1876 third edition of his *Modern Higher Algebra*: "The discriminant is equal to the product of the squares of all the differences of any two roots of the equation."<sup>[11](https://mathshistory.st-andrews.ac.uk//Miller/mathword/d/)</sup>

## Discriminants of quadratic forms and number fields

The same idea extends beyond single polynomials. For a quadratic form, the discriminant is defined up to the square of an invertible element of the base ring and is an invariant of the form's class; under a linear change of variables by a nonsingular matrix \( S \), the form's matrix becomes \( S^{\mathrm T} A S \), so the discriminant is multiplied by the square of \( \det S \).<sup>[12](https://encyclopediaofmath.org/wiki/Quadratic_form)</sup> For binary quadratic forms \( ax^2+bxy+cy^2 \) over the integers, the discriminant \( b^2-4ac \) is preserved by unimodular changes of variables, and equivalent forms represent the same integers and share the same discriminant.<sup>[13](https://crypto.stanford.edu/pbc/notes/numbertheory/form.html)</sup>

In algebraic number theory, the discriminant of a number field \( K \) of degree \( n \) is the determinant of the matrix \( (\mathrm{Tr}_{K/\mathbb{Q}}(\omega_i \omega_j)) \) built from an integral basis \( \omega_1, \dots, \omega_n \) of its ring of integers; it is a nonzero integer, and changing the basis multiplies it by the square of the change-of-basis determinant, which for integral bases is \( \pm 1 \).<sup>[5](https://websites.math.leidenuniv.nl/algebra/ant.pdf)</sup><sup> • </sup><sup>[14](https://crypto.stanford.edu/pbc/notes/numberfield/discriminant.html)</sup> Its arithmetic meaning is ramification: a rational prime divides the field discriminant exactly when it ramifies in \( K \), and the different ideal, a finer invariant living inside the ring of integers, has norm \( |\Delta_K| \).<sup>[5](https://websites.math.leidenuniv.nl/algebra/ant.pdf)</sup> The discriminant also bounds how many fields exist: for any \( N \ge 1 \), only finitely many number fields \( K \) satisfy \( |\mathrm{disc}(K)| \le N \).<sup>[15](https://www.math.ucla.edu/~sharifi/notes/algnum-ch06.html)</sup>

## Geometric and scheme-theoretic viewpoints

In algebraic geometry the discriminant appears as a locus. For a family of polynomials or hypersurfaces parametrized by some base space, the discriminant locus is the set of parameters for which a fiber acquires a singularity, for example a multiple root.<sup>[16](https://0-academic-oup-com.legcat.gov.ns.ca/qjmath/article/75/4/1301/7801179?searchresult=1)</sup> The Stacks project generalizes this to morphisms of schemes: for a finite locally free morphism \( \pi \), the discriminant is the closed subscheme cut out by a canonical global section built from the trace pairing, and \( \pi \) is étale, meaning unramified and nonsingular, if and only if its discriminant is empty.<sup>[17](https://stacks.math.columbia.edu/tag/0BVH)</sup> The complement of the discriminant locus in the space of complex polynomials of degree \( p \) has fundamental group the braid group, which records how the roots permute as the coefficients move around the locus where roots collide.<sup>[18](https://digitalcommons.cwu.edu/math/81/)</sup>

## References

1. [Discriminant, MacTutor History of Mathematics](https://mathshistory.st-andrews.ac.uk/Extras/Discriminant/)
2. [Polynomial Discriminant, Wolfram MathWorld](https://mathworld.wolfram.com/PolynomialDiscriminant.html)
3. [Introduction to Algebraic Geometry, MIT 18.721 notes (resultant and discriminant)](https://math.mit.edu/classes/18.721/notes/ag-dec1-2021.pdf)
4. [K. H. Parshall, on Sylvester's coinage of "discriminant", Revue d'histoire des mathématiques](https://smf.emath.fr/system/files/filepdf/HUNGER-PARSHALL.pdf)
5. [Algebraic Number Theory, §4 Linear algebra for number rings, Leiden University notes](https://websites.math.leidenuniv.nl/algebra/ant.pdf)
6. [Faber, Ingalls, May, Talarico (2024). Matrix factorizations of the discriminant of S_n. Journal of Symbolic Computation 125, 102310](https://eprints.whiterose.ac.uk/id/eprint/209753/1/Matrix%20factorizations%20of%20the%20discriminant%20of%20Sn.pdf)
7. [The resultant, the discriminant, and the derivative of generalized Fibonacci polynomials, arXiv:1808.01264](https://arxiv.org/abs/1808.01264)
8. [Magma Handbook: Resultants and Discriminants](https://magma.maths.usyd.edu.au/magma/handbook/text/249)
9. [James Joseph Sylvester (1814–1897), MacTutor Biography](https://mathshistory.st-andrews.ac.uk//Biographies/Sylvester/)
10. [Sylvester and invariants, ERIC document ED054941](https://files.eric.ed.gov/fulltext/ED054941.pdf)
11. [Earliest Known Uses of Some of the Words of Mathematics (D), MacTutor](https://mathshistory.st-andrews.ac.uk//Miller/mathword/d/)
12. [Quadratic form, Encyclopedia of Mathematics](https://encyclopediaofmath.org/wiki/Quadratic_form)
13. [Number Theory: Binary Quadratic Forms, Stanford notes](https://crypto.stanford.edu/pbc/notes/numbertheory/form.html)
14. [The Discriminant, Stanford number field notes](https://crypto.stanford.edu/pbc/notes/numberfield/discriminant.html)
15. [Romyar Sharifi, Algebraic Number Theory, Chapter 6: Ramification theory](https://www.math.ucla.edu/~sharifi/notes/algnum-ch06.html)
16. [Discriminant Divisors for Conic Bundles, Quarterly Journal of Mathematics](https://0-academic-oup-com.legcat.gov.ns.ca/qjmath/article/75/4/1301/7801179?searchresult=1)
17. [The Stacks Project, Section 49.3: Discriminant of a finite locally free morphism](https://stacks.math.columbia.edu/tag/0BVH)
18. [A. D. Raza Choudary, On the resultant hypersurface, Pacific Journal of Mathematics](https://digitalcommons.cwu.edu/math/81/)

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