I think that I can fully clarify the case u v 0 if u v 0 then the numerator of w is zero Otherwise a small variation of the coefficients would provide monic equations with bounded coefficients with arbitrarily large roots in absolute value this is impossible A purely algebraic proof is possible using Galois theory but I should think harder to find such an algebraic proof As the numerator of w is zero if and only if the associated depressed quartic is bi quadratic one may eliminate this case by considering separately the quartics whose depressed form is bi quadratic If one want a truly general formula one may choose another determination of the cubic root defining Q If the three determinations give u v 0 this implies that all the coefficients of the associate depressed cubic are zero to be verified but I am pretty sure thus the quartic is