0
$\begingroup$

$x=\sqrt[3]{q+\sqrt{q^2-p^3}}-\sqrt[3]{-q+\sqrt{q^2-p^3}}$

Why is it that this must be true $q^2-p^3<0$ for x to have 3 distinct real roots?

  • 0
    Yes, sorry- will adjust.2012-09-19

1 Answers 1

1

This is the casus irreducibilis of the cubic. You need to use complex numbers even if the roots are all real. The proof needs Galois theory.

  • 0
    @Alyosha, it may be. My point is that proving that you cannot solve a cubic with three real roots using radicals without using complex numbers needs Galois theory. Not that if q^2-p^3<0 then it has three real roots. I seems that I did not really answer your question. I suggest you add an answer yourself.2012-09-27