Prove irreducibility of a polynomial Let $m$ be an integer, squarefree, $m\neq 1$. Prove that $x^3-m$ is irreducible in $\mathbb{Q}[X]$.
My thoughts: since $m$ is squarefree, i have the prime factorization $m=p_1\cdots p_k$. Let $p$ be any of the primes dividing $m$. Then $p$ divides $m$, $p$ does not divide the leading coefficient, $p^2$ does not divide $m$. Hence $x^3-m$ is irreducible over $\mathbb{Q}$ by Eisenstein.
Questions:
1) Do you think it's correct?
2) Is there some different way to prove irreducibility of this polynomial.
Thanks to all.

$x^3-m$ is reducible iff it has a factor of degree 1 iff it has a root iff $m$ is a cube. In particular, $x^3-m$ is irreducible when $m$ is squarefree.
