 Solution

The elementary symmetric functions areThe Fundamental theorem of symmetric polynomials says that every symmetric polynomial is uniquely a polynomial in $e_1,\dots ,e_n$.

If $r_1,\dots ,r_n$ are the roots of a monic polynomial, its discriminant isThis is symmetric in the roots, hence the theorem and Vieta's formulas express it polynomially in the coefficients. Equivalently,For $x^5+q$ this givesFor $f=x^5+p{x}^2+q$, a repeated root solves $f=f^{\prime }=0$. Elimination givesThus the discriminant vanishes exactly when $q=0$ or $108p^5+3125q^3=0$.