Editing Discriminant (section)

==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 field]]s, 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 {{math|''A''}} be a homogeneous polynomial in {{math|''n''}} indeterminates over a field of [[characteristic (algebra)|characteristic]] 0, or of a [[prime number|prime]] characteristic that does not [[divisor|divide]] the degree of the polynomial. The polynomial {{math|''A''}} defines a [[projective hypersurface]], which has [[singular point of an algebraic variety|singular points]] if and only the {{math|''n''}} [[partial derivative]]s of {{math|''A''}} have a nontrivial common [[zero of a function|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 {{math|''A''}}. However, because of the integer coefficients resulting of the derivation, this multivariate resultant may be divisible by a power of {{math|''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 because 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 {{math|''d''}}, this general discriminant is <math>d^{d-2}</math> times the discriminant defined in {{slink||Homogeneous bivariate polynomial}}. Several other classical types of discriminants, that are instances of the general definition are described in next sections.

===Quadratic forms===
{{See also|Fundamental discriminant}}
A [[quadratic form]] is a function over a [[vector space]], which is defined over some [[basis (vector space)|basis]] by a [[homogeneous polynomial]] of degree 2:

:<math>Q(x_1,\ldots,x_n) \ =\ \sum_{i=1}^n a_{ii} x_i^2+\sum_{1\le i <j\le n}a_{ij}x_i x_j,</math>
or, in matrix form,
:<math>Q(X) =X A X^\mathrm T,</math>

for the <math>n\times n</math> [[symmetric matrix]] <math>A=(a_{ij})</math>, the <math>1\times n</math> row vector <math>X=(x_1,\ldots,x_n)</math>, and the <math>n\times 1</math> column vector <math>X^{\mathrm{T}}</math>. In [[characteristic (algebra)|characteristic]] different from 2,<ref>In characteristic 2, the discriminant of a quadratic form is not defined, and is replaced by the [[Arf invariant]].</ref> the '''discriminant''' or '''determinant''' of {{math|''Q''}} is the [[determinant]] of {{math|''A''}}.<ref>{{cite book | first=J. W. S. | last=Cassels | author-link=J. W. S. Cassels | title=Rational Quadratic Forms | series=London Mathematical Society Monographs | volume=13 | publisher=[[Academic Press]] | year=1978 | isbn=0-12-163260-1 | zbl=0395.10029 | page=6 }}</ref>

The [[Hessian determinant]] of {{math|''Q''}} is <math>2^n</math> times its discriminant. The [[multivariate resultant]] of the partial derivatives of {{math|''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]] {{math|''S''}}, changes the matrix {{math|''A''}} into <math>S^\mathrm T A\,S,</math> and thus multiplies the discriminant by the square of the determinant of {{math|''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 {{math|''K''}} is an element of {{math|''K''/(''K''<sup>×</sup>)<sup>2</sup>}}, the [[quotient monoid|quotient]] of the multiplicative [[monoid]] of {{math|''K''}} by the [[subgroup]] of the nonzero squares (that is, two elements of {{math|''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 number]]s, a discriminant is equivalent to 0 or 1. Over the [[real number]]s, a discriminant is equivalent to −1, 0, or 1. Over the [[rational number]]s, a discriminant is equivalent to a unique [[square-free integer]].

By a theorem of [[Carl Gustav Jacob Jacobi|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
:<math>a_1x_1^2 + \cdots + a_nx_n^2.</math>
More precisely, a quadratic form may be expressed as a sum
:<math>\sum_{i=1}^n a_i L_i^2</math>
where the {{math|''L''<sub>''i''</sub>}} are independent linear forms and {{mvar|n}} is the number of the variables (some of the {{math|''a''<sub>''i''</sub>}} may be zero). Equivalently, for any symmetric matrix {{math|''A''}}, there is an [[elementary matrix]] {{math|''S''}} such that <math>S^\mathrm T A\,S</math> is a [[diagonal matrix]].
Then the discriminant is the product of the {{math|''a''<sub>''i''</sub>}}, which is well-defined as a class in {{math|''K''/(''K''<sup>×</sup>)<sup>2</sup>}}.

Geometrically, the discriminant of a quadratic form in three variables is the equation of a [[projective curve|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 of an algebraic variety|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
:<math>ax^2+ 2bxy + cy^2 + 2dx + 2ey + f = 0,</math>
where {{math|''a'', ''b'', ''c'', ''d'', ''e'', ''f''}} are real numbers.

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

The first quadratic form is
:<math>ax^2+ 2bxy + cy^2 + 2dxz + 2eyz + fz^2 = 0.</math>
Its discriminant is the [[determinant]]
:<math>\begin{vmatrix} a & b & d\\b & c & e\\d & e & f \end{vmatrix}. </math>
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<ref>{{cite book
|title=Math refresher for scientists and engineers
|first1=John R.
|last1=Fanchi
|publisher=John Wiley and Sons
|year=2006
|isbn=0-471-75715-2
|url=https://books.google.com/books?id=75mAJPcAWT8C&pg=PA45
|at=sec. 3.2, p. 45}}
</ref>

:<math>b^2 - ac,</math>

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 <math>P(x,y,z)</math> be a polynomial of degree two in three variables that defines a real quadric surface. The first associated quadratic form, <math>Q_4,</math> depends on four variables, and is obtained by [[homogenization of a polynomial|homogenizing]] {{math|''P''}}; that is
:<math>Q_4(x,y,z,t)=t^2P(x/t,y/t, z/t).</math>
Let us denote its discriminant by <math>\Delta_4.</math>

The second quadratic form, <math>Q_3,</math> depends on three variables, and consists of the terms of degree two of {{math|''P''}}; that is 
:<math>Q_3(x,y,z)=Q_4(x, y,z,0).</math>
Let us denote its discriminant by <math>\Delta_3.</math>

If <math>\Delta_4>0,</math> 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 <math>\Delta_4<0,</math> 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 <math>\Delta_4=0,</math> the surface has a [[singular point of an algebraic variety|singular point]], possibly [[point at infinity|at infinity]]. If there is only one singular point, the surface is a [[cylinder]] or a [[conic surface|cone]]. If there are several singular points the surface consists of two planes, a double plane or a single line.

When <math>\Delta_4\ne 0,</math> the sign of <math>\Delta_3,</math> if not 0, does not provide any useful information, as changing {{math|''P''}} into {{math|−''P''}} does not change the surface, but changes the sign of <math>\Delta_3.</math> However, if <math>\Delta_4\ne 0</math> and <math>\Delta_3 = 0,</math> the surface is a [[paraboloid]], which is elliptic or hyperbolic, depending on the sign of <math>\Delta_4.</math>

===Discriminant of an algebraic number field===
{{main article|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 [[Prime number|primes]] are [[Ramified prime#In algebraic number theory|ramified]].

The discriminant is one of the most basic invariants of a number field, and occurs in several important [[Analytic Number Theory|analytic]] formulas such as the [[Functional equation (L-function)|functional equation]] of the [[Dedekind zeta function]] of ''K'', and the [[analytic class number formula]] for ''K''. [[Hermite–Minkowski theorem|A theorem]] of [[Charles Hermite|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.<ref>{{Citation| last1=Cohen| first1=Henri| author-link=Henri Cohen (number theorist)| last2=Diaz y Diaz| first2=Francisco | last3=Olivier| first3=Michel| contribution=A Survey of Discriminant Counting| title=Algorithmic Number Theory, Proceedings, 5th International Syposium, ANTS-V, University of Sydney, July 2002| editor-last=Fieker| editor-first=Claus| editor2-last=Kohel| editor2-first=David R.| publisher=Springer-Verlag| location=Berlin| series=Lecture Notes in Computer Science | issn=0302-9743| isbn=978-3-540-43863-2| doi=10.1007/3-540-45455-1_7| year=2002| volume=2369| pages=80–94| mr=2041075}}</ref>

Let ''K'' be an algebraic number field, and let ''O<sub>K</sub>'' be its [[ring of integers]]. Let ''b''<sub>1</sub>, ..., ''b<sub>n</sub>'' be an [[integral basis]] of ''O<sub>K</sub>'' (i.e. a basis as a [[Module (mathematics)|'''Z'''-module]]), and let {σ<sub>1</sub>, ..., σ<sub>''n''</sub>} be the set of embeddings of ''K'' into the [[Complex number|complex numbers]] (i.e. [[injective]] [[Ring homomorphism|ring homomorphisms]] ''K''&nbsp;→&nbsp;'''C'''). The '''discriminant''' of ''K'' is the [[Square (algebra)|square]] of the [[determinant]] of the ''n'' by ''n'' [[Matrix (mathematics)|matrix]] ''B'' whose (''i'',''j'')-entry is σ<sub>''i''</sub>(''b<sub>j</sub>''). Symbolically,

: <math>\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.
</math>


The discriminant of ''K'' can be referred to as the absolute discriminant of ''K'' to distinguish it from the  of an [[Field extension|extension]] ''K''/''L'' of number fields. The latter is an [[Ideal (ring theory)|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''&nbsp;=&nbsp;'''Q''', the relative discriminant of ''K''/'''Q''' is the [[principal ideal]] of '''Z''' generated by the absolute discriminant of ''K''.
===Fundamental discriminants===

A specific type of discriminant useful in the study of quadratic fields is the fundamental discriminant. It arises in the theory of integral [[Binary quadratic form|binary quadratic forms]], which are expressions of the form:<math display="block">Q(x, y) = ax^2 + bxy + cy^2</math>


where <math display="inline">a</math>, <math display="inline">b</math>, and <math display="inline">c</math> are integers. The discriminant of <math display="inline">Q(x, y)</math> is given by:<math display="block">D = b^2 - 4ac</math>Not every integer can arise as a discriminant of an integral binary quadratic form. An integer <math display="inline">D</math> is a fundamental discriminant if and only if it meets one of the following criteria:

* Case 1: <math display="inline">D</math> is congruent to 1 modulo 4 (<math display="inline">D \equiv 1 \pmod{4}</math>) and is square-free, meaning it is not divisible by the square of any prime number.
* Case 2: <math display="inline">D</math> is equal to four times an integer <math display="inline">m</math> (<math display="inline">D = 4m</math>) where <math display="inline">m</math> is congruent to 2 or 3 modulo 4 (<math display="inline">m \equiv 2, 3 \pmod{4}</math>) and is square-free.

These conditions ensure that every fundamental discriminant corresponds uniquely to a specific type of quadratic form.

The first eleven positive fundamental discriminants are:

: [[1 (number)|1]], [[5 (number)|5]], [[8 (number)|8]], [[12 (number)|12]], [[13 (number)|13]], [[17 (number)|17]], [[21 (number)|21]], [[24 (number)|24]], [[28 (number)|28]], [[29 (number)|29]], [[33 (number)|33]] (sequence A003658 in the [[On-Line Encyclopedia of Integer Sequences|OEIS]]).

The first eleven negative fundamental discriminants are:

: −3, −4, −7, −8, −11, −15, −19, −20, −23, −24, −31 (sequence A003657 in the [[On-Line Encyclopedia of Integer Sequences|OEIS]]).

==== Quadratic number fields ====
A quadratic field is a field extension of the rational numbers <math display="inline">\mathbb{Q}</math> that has degree 2. The discriminant of a quadratic field plays a role analogous to the discriminant of a quadratic form.

There exists a fundamental connection: an integer <math display="inline">D_0</math> is a fundamental discriminant if and only if:

* <math display="inline">D_0 = 1</math>, or
* <math display="inline">D_0</math> is the discriminant of a quadratic field.

For each fundamental discriminant <math display="inline">D_0 \neq 1</math>, there exists a unique (up to isomorphism) quadratic field with <math display="inline">D_0</math> as its discriminant. This connects the theory of quadratic forms and the study of quadratic fields.

==== Prime factorization ====
Fundamental discriminants can also be characterized by their prime factorization. Consider the set <math display="inline">S</math> consisting of <math>-8, 8, -4,</math> the prime numbers congruent to 1 modulo 4, and the [[additive inverse]]s of the prime numbers congruent to 3 modulo 4:<math display="block">S = \{-8, -4, 8, -3, 5, -7, -11, 13, 17, -19, ... \}</math>An integer <math display="inline">D \neq 1</math> is a fundamental discriminant if and only if it is a product of elements of <math>S</math> that are pairwise [[coprime]].{{cn|date=August 2024}}