Editing Discriminant (section)

==Definition==

Let 
:<math>A(x) = a_nx^n+a_{n-1}x^{n-1}+\cdots+a_1x+a_0</math>
be a polynomial of [[degree of a polynomial|degree]] {{math|''n''}} (this means <math>a_n\ne 0</math>), such that the coefficients <math>a_0, \ldots, a_n</math> belong to a [[field (mathematics)|field]], or, more generally, to a [[commutative ring]]. The [[resultant]] of {{math|''A''}} with its [[formal derivative|derivative]], 
:<math>A'(x) = na_nx^{n-1}+(n-1)a_{n-1}x^{n-2}+\cdots+a_1,</math> 
is a polynomial in <math>a_0, \ldots, a_n</math> with [[integer]] coefficients, which is the [[determinant]] of the [[Sylvester matrix]] of {{math|''A''}} and {{math|''A''{{void}}′}}. The nonzero entries of the first column of the Sylvester matrix are <math>a_n</math> and <math>na_n,</math> and the [[resultant]] is thus a multiple of <math>a_n.</math> Hence the discriminant—up to its sign—is defined as the quotient of the resultant of {{math|''A''}} and {{math|''A'{{void}}''}} by <math>a_n</math>:

:<math>\operatorname{Disc}_x(A) = \frac{(-1)^{n(n-1)/2}}{a_n} \operatorname{Res}_x(A,A')</math>

Historically, this sign has been chosen such that, over the reals, the discriminant will be positive when all the roots of the polynomial are real. The division by <math>a_n</math> may not be well defined if the [[ring (mathematics)|ring]] of the coefficients contains [[zero divisor]]s. Such a problem may be avoided by replacing <math>a_n</math> by 1 in the first column of the Sylvester matrix—''before'' computing the determinant. In any case, the discriminant is a polynomial in <math>a_0, \ldots, a_n</math> with integer coefficients.

===Expression in terms of the roots===
When the above polynomial is defined over a [[field (mathematics)|field]], it has {{math|''n''}} roots, <math>r_1, r_2, \dots, r_n</math>, 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 number]]s, where the [[fundamental theorem of algebra]] applies.) 

In terms of the roots, the discriminant is equal to

:<math>\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).</math>

It is thus the square of the [[Vandermonde polynomial]] times <math>a_n^{2n-2} </math>.

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 root|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 {{math|''A''}}.