Exercise 1.3.5

Proof. With the hypothesis $\mathrm{\Delta }\ne 0$, $f(y)=y^3+\mathit{py}+q$ has three distinct roots (Ex 1.2.4). We have proved in Ex. 1.3.4 that $\mathrm{\Delta }>0$ iff the three roots of $f$ are real: This is the part (a) of Theorem 1.3.1.

If there exists a non real complex root, Ex.4(d) show that $\mathrm{\Delta }<0$.

Conversely, if $\mathrm{\Delta }<0$, there exists a non real root $y_1$, and $y_2=\overline{y_1}$ is a root of $f$, with $y_2\ne y_1$. As $f$ has always a real root by Ex. 2,3,4, $f$ has so exactly one real root, and two non real conjugate roots. This is the part (b) of the theorem. □