Expression in terms of the roots

When the above polynomial is defined over a field, it has n roots, ${\displaystyle r_{1},r_{2},\dots ,r_{n}}$, not necessarily all distinct, in any algebraically closed extension of the field. (If the coefficients are real numbers, the roots may be taken in the field of complex numbers, where the fundamental theorem of algebra applies.)

In terms of the roots, the discriminant is equal to

${\displaystyle \operatorname {Disc} _{x}(A)=a_{n}^{2n-2}\prod _{i<j}(r_{i}-r_{j})^{2}=(-1)^{n(n-1)/2}a_{n}^{2n-2}\prod _{i\neq j}(r_{i}-r_{j}).}$

It is thus the square of the Vandermonde polynomial times ${\displaystyle a_{n}^{2n-2}}$.

This expression for the discriminant is often taken as a definition. It makes clear that if the polynomial has a multiple root, then its discriminant is zero, and that, in the case of real coefficients, if all the roots are real and simple, then the discriminant is positive. Unlike the previous definition, this expression is not obviously a polynomial in the coefficients, but this follows either from the fundamental theorem of Galois theory, or from the fundamental theorem of symmetric polynomials and Vieta's formulas by noting that this expression is a symmetric polynomial in the roots of A.

Degree 2

The quadratic polynomial ${\displaystyle ax^{2}+bx+c\,}$ has discriminant

${\displaystyle b^{2}-4ac\,.}$

The square root of the discriminant appears in the quadratic formula for the roots of the quadratic polynomial:

${\displaystyle x_{1,2}={\frac {-b\pm {\sqrt {b^{2}-4ac}}}{2a}}.}$

where the discriminant is zero if and only if the two roots are equal. If a, b, c are real numbers, the polynomial has two distinct real roots if the discriminant is positive, and two complex conjugate roots if it is negative.^[7]

The discriminant is the product of a² and the square of the difference of the roots.

If a, b, c are rational numbers, then the discriminant is the square of a rational number if and only if the two roots are rational numbers.

Degree 3

The zero set of discriminant of the cubic x³ + bx² + cx + d, i.e. points satisfying b²c² – 4c³ – 4b³d – 27d² + 18bcd = 0.

The cubic polynomial ${\displaystyle ax^{3}+bx^{2}+cx+d\,}$ has discriminant

${\displaystyle b^{2}c^{2}-4ac^{3}-4b^{3}d-27a^{2}d^{2}+18abcd\,.}$^[8][9]

In the special case of a depressed cubic polynomial ${\displaystyle x^{3}+px+q}$, the discriminant simplifies to

${\displaystyle -4p^{3}-27q^{2}\,.}$

The discriminant is zero if and only if at least two roots are equal. If the coefficients are real numbers, and the discriminant is not zero, the discriminant is positive if the roots are three distinct real numbers, and negative if there is one real root and two complex conjugate roots.^[10]

The square root of a quantity strongly related to the discriminant appears in the formulas for the roots of a cubic polynomial. Specifically, this quantity can be −3 times the discriminant, or its product with the square of a rational number; for example, the square of 1/18 in the case of Cardano formula.

If the polynomial is irreducible and its coefficients are rational numbers (or belong to a number field), then the discriminant is a square of a rational number (or a number from the number field) if and only if the Galois group of the cubic equation is the cyclic group of order three.

Degree 4

The discriminant of the quartic polynomial x⁴ + cx² + dx + e. The surface represents points (c, d, e) where the polynomial has a repeated root. The cuspidal edge corresponds to the polynomials with a triple root, and the self-intersection corresponds to the polynomials with two different repeated roots.

The quartic polynomial ${\displaystyle ax^{4}+bx^{3}+cx^{2}+dx+e\,}$ has discriminant

${\displaystyle {\begin{aligned}{}&256a^{3}e^{3}-192a^{2}bde^{2}-128a^{2}c^{2}e^{2}+144a^{2}cd^{2}e\\[4pt]&{}-27a^{2}d^{4}+144ab^{2}ce^{2}-6ab^{2}d^{2}e-80abc^{2}de\\[4pt]&{}+18abcd^{3}+16ac^{4}e-4ac^{3}d^{2}-27b^{4}e^{2}+18b^{3}cde\\[4pt]&{}-4b^{3}d^{3}-4b^{2}c^{3}e+b^{2}c^{2}d^{2}\,.\end{aligned}}}$

The depressed quartic polynomial ${\displaystyle x^{4}+cx^{2}+dx+e\,}$ has discriminant

${\displaystyle {\begin{aligned}{}&16c^{4}e-4c^{3}d^{2}-128c^{2}e^{2}+144cd^{2}e-27d^{4}+256e^{3}\,.\end{aligned}}}$

The discriminant is zero if and only if at least two roots are equal. If the coefficients are real numbers and the discriminant is negative, then there are two real roots and two complex conjugate roots. Conversely, if the discriminant is positive, then the roots are either all real or all non-real.

Zero discriminant

The discriminant of a polynomial over a field is zero if and only if the polynomial has a multiple root in some field extension.

The discriminant of a polynomial over an integral domain is zero if and only if the polynomial and its derivative have a non-constant common divisor.

In characteristic 0, this is equivalent to saying that the polynomial is not square-free (i.e., divisible by the square of a non-constant polynomial).

In nonzero characteristic p, the discriminant is zero if and only if the polynomial is not square-free or it has an irreducible factor which is not separable (i.e., the irreducible factor is a polynomial in ${\displaystyle x^{p}}$).

Invariance under change of the variable

The discriminant of a polynomial is, up to a scaling, invariant under any projective transformation of the variable. As a projective transformation may be decomposed into a product of translations, homotheties and inversions, this results in the following formulas for simpler transformations, where P(x) denotes a polynomial of degree n, with ${\displaystyle a_{n}}$ as leading coefficient.

• Invariance by translation:

${\displaystyle \operatorname {Disc} _{x}(P(x+\alpha ))=\operatorname {Disc} _{x}(P(x))}$

This results from the expression of the discriminant in terms of the roots

• Invariance by homothety:

${\displaystyle \operatorname {Disc} _{x}(P(\alpha x))=\alpha ^{n(n-1)}\operatorname {Disc} _{x}(P(x))}$

This results from the expression in terms of the roots, or of the quasi-homogeneity of the discriminant.

• Invariance by inversion:

${\displaystyle \operatorname {Disc} _{x}(P^{\mathrm {r} }\!\!\;(x))=\operatorname {Disc} _{x}(P(x))}$

when ${\displaystyle P(0)\neq 0.}$ Here, ${\displaystyle P^{\mathrm {r} }\!\!\;}$ denotes the reciprocal polynomial of P; that is, if ${\displaystyle P(x)=a_{n}x^{n}+\cdots +a_{0},}$ and ${\displaystyle a_{0}\neq 0,}$ then

${\displaystyle P^{\mathrm {r} }\!\!\;(x)=x^{n}P(1/x)=a_{0}x^{n}+\cdots +a_{n}.}$

Invariance under ring homomorphisms

Let ${\displaystyle \varphi \colon R\to S}$ be a homomorphism of commutative rings. Given a polynomial

${\displaystyle A=a_{n}x^{n}+a_{n-1}x^{n-1}+\cdots +a_{0}}$

in R[x], the homomorphism ${\displaystyle \varphi }$ acts on A for producing the polynomial

${\displaystyle A^{\varphi }=\varphi (a_{n})x^{n}+\varphi (a_{n-1})x^{n-1}+\cdots +\varphi (a_{0})}$

in S[x].

The discriminant is invariant under ${\displaystyle \varphi }$ in the following sense. If ${\displaystyle \varphi (a_{n})\neq 0,}$ then

${\displaystyle \operatorname {Disc} _{x}(A^{\varphi })=\varphi (\operatorname {Disc} _{x}(A)).}$

As the discriminant is defined in terms of a determinant, this property results immediately from the similar property of determinants.

If ${\displaystyle \varphi (a_{n})=0,}$ then ${\displaystyle \varphi (\operatorname {Disc} _{x}(A))}$ may be zero or not. One has, when ${\displaystyle \varphi (a_{n})=0,}$

${\displaystyle \varphi (\operatorname {Disc} _{x}(A))=\varphi (a_{n-1})^{2}\operatorname {Disc} _{x}(A^{\varphi }).}$

When one is only interested in knowing whether a discriminant is zero (as is generally the case in algebraic geometry), these properties may be summarised as:

${\displaystyle \varphi (\operatorname {Disc} _{x}(A))=0}$ if and only if either ${\displaystyle \operatorname {Disc} _{x}(A^{\varphi })=0}$ or ${\displaystyle \deg(A)-\deg(A^{\varphi })\geq 2.}$

This is often interpreted as saying that ${\displaystyle \varphi (\operatorname {Disc} _{x}(A))=0}$ if and only if ${\displaystyle A^{\varphi }}$ has a multiple root (possibly at infinity).

Product of polynomials

If R = PQ is a product of polynomials in x, then

${\displaystyle {\begin{aligned}\operatorname {disc} _{x}(R)&=\operatorname {disc} _{x}(P)\operatorname {Res} _{x}(P,Q)^{2}\operatorname {disc} _{x}(Q)\\[5pt]{}&=(-1)^{pq}\operatorname {disc} _{x}(P)\operatorname {Res} _{x}(P,Q)\operatorname {Res} _{x}(Q,P)\operatorname {disc} _{x}(Q),\end{aligned}}}$

where ${\displaystyle \operatorname {Res} _{x}}$ denotes the resultant with respect to the variable x, and p and q are the respective degrees of P and Q.

This property follows immediately by substituting the expression for the resultant, and the discriminant, in terms of the roots of the respective polynomials.

Homogeneity

The discriminant is a homogeneous polynomial in the coefficients; it is also a homogeneous polynomial in the roots and thus quasi-homogeneous in the coefficients.

The discriminant of a polynomial of degree n is homogeneous of degree 2n − 2 in the coefficients. This can be seen in two ways. In terms of the roots-and-leading-term formula, multiplying all the coefficients by λ does not change the roots, but multiplies the leading term by λ. In terms of its expression as a determinant of a (2n − 1) × (2n − 1) matrix (the Sylvester matrix) divided by a_n, the determinant is homogeneous of degree 2n − 1 in the entries, and dividing by a_n makes the degree 2n − 2.

The discriminant of a polynomial of degree n is homogeneous of degree n(n − 1) in the roots. This follows from the expression of the discriminant in terms of the roots, which is the product of a constant and ${\displaystyle {\binom {n}{2}}={\frac {n(n-1)}{2}}}$ squared differences of roots.

The discriminant of a polynomial of degree n is quasi-homogeneous of degree n(n − 1) in the coefficients, if, for every i, the coefficient of ${\displaystyle x^{i}}$ is given the weight n − i. It is also quasi-homogeneous of the same degree, if, for every i, the coefficient of ${\displaystyle x^{i}}$ is given the weight i. This is a consequence of the general fact that every polynomial which is homogeneous and symmetric in the roots may be expressed as a quasi-homogeneous polynomial in the elementary symmetric functions of the roots.

Consider the polynomial

${\displaystyle P=a_{n}x^{n}+a_{n-1}x^{n-1}+\cdots +a_{0}.}$

It follows from what precedes that the exponents in every monomial ${\displaystyle a_{0}^{i_{0}},\dots ,a_{n}^{i_{n}}}$ appearing in the discriminant satisfy the two equations

${\displaystyle i_{0}+i_{1}+\cdots +i_{n}=2n-2}$

and

${\displaystyle i_{1}+2i_{2}+\cdots +ni_{n}=n(n-1),}$

and also the equation

${\displaystyle ni_{0}+(n-1)i_{1}+\cdots +i_{n-1}=n(n-1),}$

which is obtained by subtracting the second equation from the first one multiplied by n.

This restricts the possible terms in the discriminant. For the general quadratic polynomial there are only two possibilities and two terms in the discriminant, while the general homogeneous polynomial of degree two in three variables has 6 terms. For the general cubic polynomial, there are five possibilities and five terms in the discriminant, while the general homogeneous polynomial of degree 4 in 5 variables has 70 terms.

For higher degrees, there may be monomials which satisfy above equations and do not appear in the discriminant. The first example is for the quartic polynomial ${\displaystyle ax^{4}+bx^{3}+cx^{2}+dx+e}$, in which case the monomial ${\displaystyle bc^{4}d}$ satisfies the equations without appearing in the discriminant.

Quadratic forms

A quadratic form is a function over a vector space, which is defined over some basis by a homogeneous polynomial of degree 2:

${\displaystyle Q(x_{1},\ldots ,x_{n})\ =\ \sum _{i=1}^{n}a_{ii}x_{i}^{2}+\sum _{1\leq i<j\leq n}a_{ij}x_{i}x_{j},}$

or, in matrix form,

${\displaystyle Q(X)=XAX^{\mathrm {T} },}$

for the ${\displaystyle n\times n}$ symmetric matrix ${\displaystyle A=(a_{ij})}$, the ${\displaystyle 1\times n}$ row vector ${\displaystyle X=(x_{1},\ldots ,x_{n})}$, and the ${\displaystyle n\times 1}$ column vector ${\displaystyle X^{\mathrm {T} }}$. In characteristic different from 2,^[11] the discriminant or determinant of Q is the determinant of A.^[12]

The Hessian determinant of Q is ${\displaystyle 2^{n}}$ times its discriminant. The multivariate resultant of the partial derivatives of Q is equal to its Hessian determinant. So, the discriminant of a quadratic form is a special case of the above general definition of a discriminant.

The discriminant of a quadratic form is invariant under linear changes of variables (that is a change of basis of the vector space on which the quadratic form is defined) in the following sense: a linear change of variables is defined by a nonsingular matrix S, changes the matrix A into ${\displaystyle S^{\mathrm {T} }A\,S,}$ and thus multiplies the discriminant by the square of the determinant of S. Thus the discriminant is well defined only up to the multiplication by a square. In other words, the discriminant of a quadratic form over a field K is an element of K/(K^×)², the quotient of the multiplicative monoid of K by the subgroup of the nonzero squares (that is, two elements of K are in the same equivalence class if one is the product of the other by a nonzero square). It follows that over the complex numbers, a discriminant is equivalent to 0 or 1. Over the real numbers, a discriminant is equivalent to −1, 0, or 1. Over the rational numbers, a discriminant is equivalent to a unique square-free integer.

By a theorem of Jacobi, a quadratic form over a field of characteristic different from 2 can be expressed, after a linear change of variables, in diagonal form as

${\displaystyle a_{1}x_{1}^{2}+\cdots +a_{n}x_{n}^{2}.}$

More precisely, a quadratic forms on may be expressed as a sum

${\displaystyle \sum _{i=1}^{n}a_{i}L_{i}^{2}}$

where the L_i are independent linear forms and n is the number of the variables (some of the a_i may be zero). Equivalently, for any symmetric matrix A, there is an elementary matrix S such that ${\displaystyle S^{\mathrm {T} }A\,S}$ is a diagonal matrix. Then the discriminant is the product of the a_i, which is well-defined as a class in K/(K^×)².

Geometrically, the discriminant of a quadratic form in three variables is the equation of a quadratic projective curve. The discriminant is zero if and only if the curve is decomposed in lines (possibly over an algebraically closed extension of the field).

A quadratic form in four variables is the equation of a projective surface. The surface has a singular point if and only its discriminant is zero. In this case, either the surface may be decomposed in planes, or it has a unique singular point, and is a cone or a cylinder. Over the reals, if the discriminant is positive, then the surface either has no real point or has everywhere a negative Gaussian curvature. If the discriminant is negative, the surface has real points, and has a negative Gaussian curvature.

Conic sections

A conic section is a plane curve defined by an implicit equation of the form

${\displaystyle ax^{2}+2bxy+cy^{2}+2dx+2ey+f=0,}$

where a, b, c, d, e, f are real numbers.

Two quadratic forms, and thus two discriminants may be associated to a conic section.

The first quadratic form is

${\displaystyle ax^{2}+2bxy+cy^{2}+2dxz+2eyz+fz^{2}=0.}$

Its discriminant is the determinant

${\displaystyle {\begin{vmatrix}a&b&d\\b&c&e\\d&e&f\end{vmatrix}}.}$

It is zero if the conic section degenerates into two lines, a double line or a single point.

The second discriminant, which is the only one that is considered in many elementary textbooks, is the discriminant of the homogeneous part of degree two of the equation. It is equal to^[13]

${\displaystyle b^{2}-ac,}$

and determines the shape of the conic section. If this discriminant is negative, the curve either has no real points, or is an ellipse or a circle, or, if degenerated, is reduced to a single point. If the discriminant is zero, the curve is a parabola, or, if degenerated, a double line or two parallel lines. If the discriminant is positive, the curve is a hyperbola, or, if degenerated, a pair of intersecting lines.

Real quadric surfaces

A real quadric surface in the Euclidean space of dimension three is a surface that may be defined as the zeros of a polynomial of degree two in three variables. As for the conic sections there are two discriminants that may be naturally defined. Both are useful for getting information on the nature of a quadric surface.

Let ${\displaystyle P(x,y,z)}$ be a polynomial of degree two in three variables that defines a real quadric surface. The first associated quadratic form, ${\displaystyle Q_{4},}$ depends on four variables, and is obtained by homogenizing P; that is

${\displaystyle Q_{4}(x,y,z,t)=t^{2}P(x/t,y/t,z/t).}$

Let us denote its discriminant by ${\displaystyle \Delta _{4}.}$

The second quadratic form, ${\displaystyle Q_{3},}$ depends on three variables, and consists of the terms of degree two of P; that is

${\displaystyle Q_{3}(x,y,z)=Q_{4}(x,y,z,0).}$

Let us denote its discriminant by ${\displaystyle \Delta _{3}.}$

If ${\displaystyle \Delta _{4}>0,}$ and the surface has real points, it is either a hyperbolic paraboloid or a one-sheet hyperboloid. In both cases, this is a ruled surface that has a negative Gaussian curvature at every point.

If ${\displaystyle \Delta _{4}<0,}$ the surface is either an ellipsoid or a two-sheet hyperboloid or an elliptic paraboloid. In all cases, it has a positive Gaussian curvature at every point.

If ${\displaystyle \Delta _{4}=0,}$ the surface has a singular point, possibly at infinity. If there is only one singular point, the surface is a cylinder or a cone. If there are several singular points the surface consists of two planes, a double plane or a single line.

When ${\displaystyle \Delta _{4}\neq 0,}$ the sign of ${\displaystyle \Delta _{3},}$ if not 0, does not provide any useful information, as changing P into −P does not change the surface, but changes the sign of ${\displaystyle \Delta _{3}.}$ However, if ${\displaystyle \Delta _{4}\neq 0}$ and ${\displaystyle \Delta _{3}=0,}$ the surface is a paraboloid, which is elliptic or hyperbolic, depending on the sign of ${\displaystyle \Delta _{4}.}$

Discriminant of an algebraic number field

The discriminant of an algebraic number field measures the size of the (ring of integers of the) algebraic number field.

More specifically, it is proportional to the squared volume of the fundamental domain of the ring of integers, and it regulates which primes are ramified.

The discriminant is one of the most basic invariants of a number field, and occurs in several important analytic formulas such as the functional equation of the Dedekind zeta function of K, and the analytic class number formula for K. A theorem of Hermite states that there are only finitely many number fields of bounded discriminant, however determining this quantity is still an open problem, and the subject of current research.^[14]

Let K be an algebraic number field, and let O_K be its ring of integers. Let b₁, ..., b_n be an integral basis of O_K (i.e. a basis as a Z-module), and let {σ₁, ..., σ_n} be the set of embeddings of K into the complex numbers (i.e. injective ring homomorphisms K → C). The discriminant of K is the square of the determinant of the n by n matrix B whose (i,j)-entry is σ_i(b_j). Symbolically,

${\displaystyle \Delta _{K}=\det \left({\begin{array}{cccc}\sigma _{1}(b_{1})&\sigma _{1}(b_{2})&\cdots &\sigma _{1}(b_{n})\\\sigma _{2}(b_{1})&\ddots &&\vdots \\\vdots &&\ddots &\vdots \\\sigma _{n}(b_{1})&\cdots &\cdots &\sigma _{n}(b_{n})\end{array}}\right)^{2}.}$

The discriminant of K can be referred to as the absolute discriminant of K to distinguish it from the of an extension K/L of number fields. The latter is an ideal in the ring of integers of L, and like the absolute discriminant it indicates which primes are ramified in K/L. It is a generalization of the absolute discriminant allowing for L to be bigger than Q; in fact, when L = Q, the relative discriminant of K/Q is the principal ideal of Z generated by the absolute discriminant of K.