Editing Discriminant (section)

===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.