The techniques I know right now to show a polynomial f(x) in Q[x] is irreducible are to show it is irreducible in some $F_p$ with p a prime f(x) has the same degree in $F_p$ as in Q[x], and the Eisenstein criteria.
I know that $6x^3-3x-18$ is irreducible in Q[x], but I don't see how to show it. The prime divisors of 18 are 2 and 3, both of which divide 6 so a simple Eisenstein doesn't work. And I've tried roots of the polynomial in $F_p$ for many p's, but it always seems to have roots. I'm guessing there is a simple way to solve this problem, does anyone know?
Best Answer
If a cubic polynomial is reducible, it has a linear factor, which corresponds to a root, which your polynomial does not have in $\mathbb Q$ by the rational root theorem.
Alternatively, you could show that your polynomial has no root in $\Bbb F_{11}$.