Reducibility of $2x^{100} +9x^5 −12x^3 +30x+15$ over $\mathbb{Q}$

Question

By Eisenstein's criterion, $2x^{100} +9x^5 −12x^3 +30x+15$ is irreducible over $\mathbb{Z}$. But how can we determine if it's also irreducible in $\mathbb{Q}$?

Answer 1

The Gauss Lemma implies that a polynomial $f\in \mathbb{Z}[x]$, which is irreducible over $\mathbb{Z}$, is also irreducible over $\mathbb{Q}$. The converse need not be true, as the example $f(x)=2x^2+4$ shows. This is reducible over the integers, because we can factor it as $2(x^2+2)$ and both $2$ and $x^2+2$ are not units.

Answer 2

It should follow directly from Gauss' Lemma.

The second result states that if a non-constant polynomial with integer coefficients is irreducible over the integers, then it is also irreducible if it is considered as a polynomial over the rationals (from wiki).