Can $X^5-X-1$ and $X^3+aX+b \in \mathbf{Q}[X]$ possibly have common complex roots?

Question

Guys I come across this problem in Kostrikin's basic algebra book.

I already know that $X^5-X-1$ is irreducible on $\mathbf{Q}$, and it has only one real (and irrational) root $\alpha \in (1,1.5)$. We thus have $X^5-X-1=(X-\alpha)(X^2+\beta_1X+\gamma_1)(X^2+\beta_2X+\gamma_2)$, where $\beta_i$ and $\gamma_i$ cannot be both rational. If $X^3+aX+b$ had the same complex root, we must have $X^3+aX+b=(X-\beta)(X^2+\beta X+\gamma)$. Thus $a=\gamma-\beta^2$, $b=-\beta\gamma$.

That's where I find trouble to go further. What should be done to demonstrate the possibility of taking $a,b$ to satisfy the $\beta,\gamma$?

Answer 1

If the two polynomials have a common complex root $\alpha$, then the minimal polynomial of $\alpha$ over $\Bbb{Q}$ divides both polynomials. But as you note $X^5-X-1$ is irreducible over $\Bbb{Q}$, so this is impossible.