Irreducible polynomial of degree 3 over rational numbers is separable or not

Question

As for an exercise it would be nice to know if any monic degree 3 irreducible polynomial $f\in\mathbb{Q}[X]$ is separable or not. A monic degree 2 irreducible polynomial in $\mathbb{Q}[X]$ is separable as the other case leads into contradiction.

My attempt is to assume that $f = (X - \alpha)^2(X - \beta)$ for $\alpha,\beta\in\mathbb{C}$.

The first observation is that we need $\alpha\in\mathbb{R}$ as in the other case the second root $\beta = \bar{\alpha}$ is the complex conjugate of $\alpha$ thus $f = X^3-(2\alpha+\bar{\alpha})X^2+(2\alpha\bar{\alpha}+\alpha^2)X-\alpha^2\bar{\alpha}\in\mathbb{Q}[X]$ implies that $2\alpha+\bar{\alpha}\in\mathbb{Q}$ thus $\alpha\in\mathbb{Q}$ which is a contradiction to the irreducibility of $f$.

The second observation is that $\beta\in\mathbb{R}\setminus\mathbb{Q}$.

Thus we have the following condition:

$f=X^3-(2\alpha+\beta)X^2+(2\alpha+\alpha^2)X-\alpha^2\beta\in\mathbb{Q}[X]$.

This is now where I am stuck. Can I get a contradiction from this? If yes, how? If no, what counter-example is there?

Answer 1

every irreducible polynomial over $\mathbb{Q}$ is separable since its characteristic is zero

Answer 2

If $p(X)$ has a double root $\alpha$, then $\alpha$ will also be a root of $p'(X)$. Therefore, $p(X)$ and $p'(X)$ are note co-prime and so, since $\deg p(X)=3$ and $\deg p'(X)=2$, $p(X)$ cannot be irreducible.