Editing Discriminant (section)

===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''}}.