discriminant as a product of pairwise differences of roots

Question

It is known that the discriminant of a polynomial $f \in K[x]$ can be described as a function of its coefficients via polynomial resultants. But in the explicit formulation of the determinant $\Delta$, namely as a product of the differences of the polynomial's roots $r_i$ $$ \Delta = \prod_{i<j} (r_i - r_j)^2 $$ Is there a way to see that $\Delta \in K$?

Answer 1

$\Delta$ is invariant with respect to any permutation of the roots $r_i$ (this is why we square the pairwise differencies: otherwise, $\Delta$ could change its sign after permuting the roots). Thus, it can be expressed as a polynomial of the elementary symmetric polynomials of $r_i$, which are precisely the coefficients of $f$.