Sufficient conditions for polynomial reducibility

Question

Are there any nice conditions under which the polynomial over $\mathbb{Q}$ or $\mathbb{F_p}$ is reducible (completely reducible)?

Is there a way to explain that such a condition does not exist? (which I believe is true)

Answer 1

Over $\mathbb{Q}$. If a polynomial $f(x)$ has a root $ a \in \mathbb{Q}$ then $f(x)$ is reducible. If $a$ is a comlex root of $f(x)$ and $a+\overline{a}, a \cdot \overline{a} \in \mathbb{Q}$ then $f(x)$ is reducible over $\mathbb{Q}$

Answer 2

Over $\mathbb{Q}$ you are asking whether the polynomial factors into linear factors (I think). Since there are bounds for the roots (or in general, coefficients of factors) in terms of the coefficients. So, if you combine this with Gauss' lemma there are only finitely many cases to check (for low degree polynomials, not so many, though I would not want to do this by hand).