Let consider $P\in \mathbb{Z}[X]$ a monic polynomial of degree $3$ and of the form : $P=(X-x_1)(X-x_2)(X-x_3)$
First I will prove that $(x_1+x_2)(x_1+x_3)(x_2+x_3)\in \mathbb{Z}$.
I develop : $P=(X-x_1)(X-x_2)(X-x_3)=X^3-(x_1+x_2+x_3)X^2+(x_1x_2+x_1x_3+x_2x_3)X-x_1x_2x_3$
Using elementary symmetric functions I can write that : $P=X^3-\sigma_1X^2+\sigma_2X-\sigma_3$.
By developing and replacing : $(x_1+x_2)(x_1+x_3)(x_2+x_3)=\sigma_1\sigma_2-\sigma_3$.
As $P\in \mathbb{Z}[X]$ it means that $\sigma_{1,2,3}\in \mathbb{Z}$. That's why $(x_1+x_2)(x_1+x_3)(x_2+x_3)\in \mathbb{Z}$.
Now I must deduce that the discriminant $D$ of $P$ is $\equiv 0,\ 1 \pmod{4}$ but I don't see how to proceed.
Thanks in advance !