Discriminant

In algebra, the discriminant of a polynomial is a polynomial function of its coefficients, which allows deducing some properties of the roots without computing them. For example, the discriminant of the quadratic polynomial $ax^{2}+bx+c$ is $b^{2}-4ac,$ which is zero if and only if the polynomial has a double root, and (in the case of real coefficients) is positive if and only if the polynomial has two real roots.

More generally, for a polynomial of an arbitrary degree, the discriminant is zero if and only if it has a multiple root, and, in the case of real coefficients, it is positive if and only if the number of non-real roots is a multiple of 4.

The discriminant is widely used in number theory, either directly or through its generalization as the discriminant of a number field. For factoring a polynomial with integer coefficients, the standard method consists in factoring first its reduction modulo a prime number not dividing the discriminant (and not dividing the leading coefficient). In algebraic geometry, the discriminant with respect to one of the variables characterizes the points of a hypersurface where the implicit function theorem does not apply.

The term "discriminant" was coined in 1851 by the British mathematician James Joseph Sylvester.^[1]

Definition

Let

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

be a polynomial of degree $n$ (this means $a_{n}\neq 0$ ), such that the coefficients $a_{0},\ldots ,a_{n}$ belong to a field, or, more generally, to a commutative ring. The resultant of $A$ and its derivative $A'(x)=na_{n}x^{n-1}+(n-1)a_{n-1}x^{n-2}+\cdots +a_{1}$ is a polynomial in $a_{0},\ldots ,a_{n}$ with integer coefficients, which is the determinant of the Sylvester matrix of $A$ and $A'$ . The nonzero entries of the first column of the Sylvester matrix are $a_{n}$ and $na_{n},$ and the resultant is thus a multiple of $a_{n}.$ So, the discriminant, up to its sign, is defined as the quotient of the resultant of $A$ and $A'$ by $a_{n}:$

\operatorname {Discr} _{x}(A)={\frac {(-1)^{\frac {n(n-1)}{2}}}{a_{n}}}\operatorname {Res} _{x}(A,A')

This sign has been historically chosen, in order that, over the reals, the discriminant is positive if all the roots of the polynomial are real. The division by $a_{n}$ may be not well defined if the ring of the coefficients contains zero divisors. Such a problem may be avoided by replacing $a_{n}$ by 1 in the first column of the Sylvester matrix before computing the determinant. In any case, the discriminant is a polynomial in $a_{0},\ldots ,a_{n}$ with integer coefficients.

Expression in terms of the roots

When the polynomial is defined over a field, the fundamental theorem of algebra implies that it has $n$ roots, $r 1, r 2, ..., r n$ , not necessarily all distinct, in an algebraically closed extension of the field (for a polynomial with real coefficients, this algebraically closed extension is generally chosen as the field of complex numbers).

In terms of the roots, the discriminant is equal to

\operatorname {Disc} _{x}(A)={a_{n}}^{2n-2}\prod _{i<j}{\left(r_{i}-r_{j}\right)^{2}}=(-1)^{{\frac {1}{2}}n(n-1)}{a_{n}}^{2n-2}\prod _{i\neq j}{\left(r_{i}-r_{j}\right)}

It is thus the square of the Vandermonde polynomial times $a n 2 n - 2$ .

This expression of the discriminant is often taken as a definition. It makes immediate that if the polynomial has a multiple root, then its discriminant is zero, and that, over the reals, if all the roots are real, then the discriminant is positive.

Low degrees

The discriminant of a linear polynomial (degree 1) is rarely considered. If needed, it is commonly defined to be equal to 1 (this is compatible with the usual conventions for the empty product and the determinant of the empty matrix). There is no common convention for the discriminant of a constant polynomial (degree 0).

For small degrees, the discriminant is rather simple (see below), but for higher degrees, the discriminant becomes significantly longer. The discriminant of a general quartic has 16 terms,^[2] that of a quintic has 59 terms,^[3] and that of a sextic has 246 terms.^[4] This is OEIS sequence A007878.

Degree 2

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

b^{2}-4ac\,.

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

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

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. ^[5]

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

x 3 + bx 2 + cx + d

, i.e. points satisfying

b 2 c 2 - 4 c 3 - 4 b 3 d - 27 d 2 + 18 bcd = 0

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

b^{2}c^{2}-4ac^{3}-4b^{3}d-27a^{2}d^{2}+18abcd\,.

In particular, the polynomial $x^{3}+px+q$ has discriminant

-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. ^[6]

The square root of the product of the discriminant by $-3$ (and possibly also by the square of a rational number) appears in the formulas for the roots of a cubic polynomial.

If the coefficients are rational numbers (or belong to a number field), then the discriminant is a square 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

x 4 + cx 2 + 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 $ax^4+bx^3+cx^2+dx+e\,$ has discriminant

{\begin{aligned}{}&256a^{3}e^{3}-192a^{2}bde^{2}-128a^{2}c^{2}e^{2}+144a^{2}cd^{2}e\\&{}-27a^{2}d^{4}+144ab^{2}ce^{2}-6ab^{2}d^{2}e-80abc^{2}de\\&{}+18abcd^{3}+16ac^{4}e-4ac^{3}d^{2}-27b^{4}e^{2}+18b^{3}cde\\&{}-4b^{3}d^{3}-4b^{2}c^{3}e+b^{2}c^{2}d^{2}\,.\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 non-zero, the discriminant is negative if there are two real roots and two complex conjugate roots, and it is positive if the roots are either all real or all non-real.

Properties

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 (that is, the polynomial is 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 not separable (that is, the irreducible factor is a polynomial in $x^{p}$ ).

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 of the following formulas for simpler transformations, where $P (x)$ denotes a polynomial in the variable $x$ of degree $n$ , with $a_{n}$ as leading coefficient.

Invariance by translation:

\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:

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

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

Invariance by inversion:

\operatorname {Disc} _{x}(P^{r}(x))=\operatorname {Disc} _{x}(P(x))

Here

P^{r}

denotes the reciprocal polynomial of

P

. That is, if

P(x)=a_{n}x^{n}+\cdots +a_{0},

then

P^{r}(x)=x^{n}P(1/x)=a_{0}x^{n}+\cdots +a_{n}.

Invariance under ring homomorphisms

Let $\varphi \colon R\to S$ be an homomorphism of commutative ring. Given a polynomial

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

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

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 $\varphi$ in the following sense. If $\varphi (a_{n})\neq 0,$ then

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

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

When, as it is generally the case in algebraic geometry, one is interested only to know if a discriminant is zero or not, these properties may be summarised into:

\varphi (\operatorname {Disc} _{x}(A))=0

if and only either

\operatorname {Disc} _{x}(A^{\varphi })=0

\deg(A)-\deg(A^{\varphi })\geq 2.

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

Product of polynomials

If $P = QR$ is a product of polynomials in $x$ , then

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

where $Res()$ denotes the resultant, 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 $2 n - 2$ in the coefficients. This can be seen 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 $(2 n - 1) \times (2 n - 1)$ matrix (the Sylvester matrix) divided by $a n$ , the determinant is homogeneous of degree $2 n - 1$ in the entries, and dividing by $a n$ makes the degree $2 n - 2$ .

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 ${\binom {n}{2}}={\frac {n(n-1)}{2}}$ squared differences of roots.

The discriminant of a polynomial of degree $n$ is quasi-homogeneous of degree $n (n - 1$ in the coefficient, if, for every $i$ , the coefficient of $x^{i}$ is given the weight $n - i$ . It is also quasi-homogeneous of the same degree, if, for every $i$ , the coefficient of $x^{i}$ 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.

Let us consider the polynomial

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 $a 0 i 0 . ..., a n i n$ appearing in the discriminant satisfies the two equations

i_0+i_1+\cdots+i_n=2n-2

and

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

and also the equation

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

which is the difference of these two equations.

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 monomials which satisfy above equations and do not appear in the discriminant. The first example is for the quartic polynomial $ax 4 + bx 3 + cx 2 + dx + e$ , in which case the monomial $bc 4 d$ satisfies the equations without appearing in the discriminant.

Real roots

In this section, all polynomials have real coefficients.

It has been seen in § Low degrees that the sign of the discriminant provides a full information on the nature of the roots for polynomials of degree 2 and 3. For higher degrees, the information provided by the discriminant is less complete, but still useful. More precisely, for a polynomial of degree $n$ , one has:

The polynomial has a multiple root if and only if its discriminant is zero.
If the discriminant is positive, the number of non-real roots is a multiple of 4. That is, there is a nonnegative integer $k \leq n /4$ such there are $2 k$ pairs of complex conjugate roots and $n - 4 k$ real roots.
If the discriminant is negative, the number of non-real roots is not a multiple of 4. That is, there is a nonnegative integer $k \leq (n - 2)/4$ such there are $2 k + 1$ pairs of complex conjugate roots and $n - 4 k + 2$ real roots.

Homogeneous bivariate polynomial

Let

A(x,y)=a_{0}x^{n}+a_{1}x^{n-1}y+\cdots +a_{n}y^{n}=\sum _{i=0}^{n}a_{i}x^{n-i}y^{i}

be a homogeneous polynomial of degree $n$ in two indeterminates.

Supposing, for the moment, that $a_{0}$ and $a_{n}$ are both nonzero, one has

\operatorname {Disc} _{x}(A(x,1))=\operatorname {Disc} _{y}(A(1,y)).

Denoting this quantity by $\operatorname {Disc} ^{h}(A),$ one has

\operatorname {Disc} _{x}(A)=y^{n(n-1)}\operatorname {Disc} ^{h}(A),

and

\operatorname {Disc} _{y}(A)=x^{n(n-1)}\operatorname {Disc} ^{h}(A).

Because of these properties, the quantity $\operatorname {Disc} ^{h}(A)$ is called the discriminant or the homogeneous discriminant of $A$ .

If $a_{0}$ and $a_{n}$ may be zero, the polynomials $A (x, 1)$ and $A (1, y)$ may have a degree smaller than $n$ . In this case, above formulas and definition remain valid, if the discriminants are computed as if all polynomials would have the degree $n$ . This means that the discriminants must be computed with $a_{0}$ and $a_{n}$ indeterminate, the substitution for them of their actual values being done after this computation. Equivalently, the formulas of § Invariance under ring homomorphisms must be used.

Use in algebraic geometry

The typical use of discriminants in algebraic geometry is for studying algebraic curve and, more generally algebraic hypersurfaces. Let $V$ be such a curve or hypersurface; $V$ is defined as the zero set of a multivariate polynomial. This polynomial may be considered as a univariate polynomial in one of the indeterminates, with polynomials in the other indeterminates as coefficients. The discriminant with respect to the selected indeterminate defines a hypersurface $W$ in the space of the other indeterminates. The points of $W$ are exactly the projection of the points of $V$ (including the points at infinity), which either are singular or have a tangent hyperplane that is parallel to the axis of the selected indeterminate.

For example, let $f$ be a bivariate polynomial in $X$ and $Y$ with real coefficients, such that $f = 0$ is the implicit equation of a plane algebraic curve. Viewing $f$ as a univariate polynomial in $Y$ with coefficients depending on $X$ , then the discriminant is a polynomial in $X$ whose roots are the $X$ -coordinates of the singular points, of the points with a tangent parallel to the $Y$ -axis and of some of the asymptotes parallel to the $Y$ -axis. In other words, the computation of the roots of the $Y$ -discriminant and the $X$ -discriminant allows one to compute all of the remarkable points of the curve, except the inflection points.

Generalizations

There are two classes of the concept of discriminant. The first class is the discriminant of an algebraic number field, which, in some cases including quadratic fields, is the discriminant of a polynomial defining the field.

Discriminants of the second class arise for problems depending on coefficients, when degenerate instances or singularities of the problem are characterized by the vanishing of a single polynomial in the coefficients. This is the case for the discriminant of a polynomial, which is zero when two roots collapse. Most of the cases, where such a generalized discriminant is defined, are instances of the following.

Let $A$ be a homogeneous polynomial in $n$ indeterminates over a field of characteristic 0, or of a prime characteristic that does not divides the degree of the polynomial. The polynomial $A$ defines a projective hypersurface, which has singular points if and only the $n$ partial derivatives of $A$ have a nontrivial common zero. This is the case if and only if the multivariate resultant of these partial derivatives is zero, and this resultant may be considered as the discriminant of $A$ . However, because of the integer coefficients resulting of the derivation, this multivariate resultant may be divisible by a power of $n$ , and it is better to take, as a discriminant, the primitive part of the resultant, computed with generic coefficients. The restriction on the characteristic is needed, as, otherwise, a common zero of the partial derivative is not necessarily a zero of the polynomial (see Euler's identity for homogeneous polynomials).

In the case of a homogeneous bivariate polynomial of degree $d$ , this general discriminant is $d^{d-2}$ times the discriminant defined in § Homogeneous bivariate polynomial. Several other classical types of discriminants, that are instances of the general definition are described in next sections.

Quadratic forms

A quadratic form is function over a vector space, which is defined over some basis by a homogeneous polynomial of degree 2. In characteristic different of 2,^[7] it may be written, w.l.o.g.

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

or, in matrix form,

Q=X^{\mathrm {T} }AX,

where $A$ is the symmetric matrix that has $a_{i,j}$ as the coefficient of the $i th$ row and $j$ th column, and $X^{\mathrm {T} }$ and $X$ are, respectively, the column vector and the row vector of the $x_{i}.$

The discriminant or determinant of $Q$ is the determinant of $A$ .^[8]

The Hessian determinant of $Q$ is $2^{n}$ 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 $S^{\mathrm {T} }A\,S,$ 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 $K /(K \times) 2$ , 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

a_1x_1^2 + \cdots + a_nx_n^2.

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

\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 $S^{\mathrm {T} }A\,S$ is a diagonal matrix. Then the discriminant is the product of the $a i$ , which is well-defined as a class in $K /(K \times) 2$ .

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 variable 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

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

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

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

The first quadratic form is

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

Its discriminant is the determinant

{\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^[9]

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

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

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

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

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

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

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

Discriminant of an algebraic number field

References

↑ Sylvester, J. J. (1851). "On a remarkable discovery in the theory of canonical forms and of hyperdeterminants". Philosophical Magazine. 4th series. 2: 391–410.
Sylvester coins the word "discriminant" on page 406.
↑ Wang, Dongming (2004). Elimination practice: software tools and applications. Imperial College Press. ch. 10 p. 180. ISBN 1-86094-438-8.
↑ Gelfand, I. M.; Kapranov, M. M.; Zelevinsky, A. V. (1994). Discriminants, resultants and multidimensional determinants. Birkhäuser. p. 1. ISBN 3-7643-3660-9.
↑ Dickenstein, Alicia; Emiris, Ioannis Z. (2005). Solving polynomial equations: foundations, algorithms, and applications. Springer. ch. 1 p. 26. ISBN 3-540-24326-7.
↑ Irving, Ronald S. (2004). Integers, polynomials, and rings. Springer-Verlag New York, Inc. ch. 10.3 pp. 153–154. ISBN 0-387-40397-3.
↑ Irving, Ronald S. (2004). Integers, polynomials, and rings. Springer-Verlag New York, Inc. ch. 10 ex. 10.14.4 & 10.17.4, pp. 154–156. ISBN 0-387-40397-3.
↑ In characteristic 2, the discriminant of a quadratic form is not defined, and is replaced by the Arf invariant.
↑ Cassels, J. W. S. (1978). Rational Quadratic Forms. London Mathematical Society Monographs. 13. Academic Press. p. 6. ISBN 0-12-163260-1. Zbl 0395.10029.
↑ Fanchi, John R. (2006). Math refresher for scientists and engineers. John Wiley and Sons. sec. 3.2, p. 45. ISBN 0-471-75715-2.

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