We know that the coefficient of ${X}^{d}$ of $F$ is positive, and that all zeros of $F(X,1)$ are in $\mathbb{C}\mathrm{\setminus}\mathbb{R}$. We also know the discriminant of $F$ is negative.

I have trouble finding this out by myself. I also looked up some number theory books, but most literature is about binary quadratic forms, not the binary form I have to prove this for. Hope someone can help me!