Gauss' Lemma

Lemma: A polynomial in $\mathbb{Z}[x]$ is irreducible if and only if it is irreducible over $\mathbb{Q}[x]$.

Proof: Let $m,n$ be the gcd’s of the coefficients of $f,g \in \mathbb{Z}[x]$. Then $m n$ divides the gcd of the coefficients of $f g$. We wish to show that this is in fact an equality.

Divide $f$ by $m$ and $g$ by $n$, so that we need only consider the case $m = n = 1$. It suffices to show that the gcd $d$ of the coefficients of $f g$ must be 1. If $d\gt 1$, then let $p$ be some prime dividing $d$. Consider the equation $f g = 0 \pmod{p}$. Since $\mathbb{Z}_p[x]$ is an integral domain, this means $f = 0$ or $g=0$, implying that $p$ divides all the coefficients of $f$ or $g$, which is a contradiction. Thus $d = 1$.

Now suppose $f = g h$ over $\mathbb{Q}[x]$. Find $m,n \in \mathbb{Q}$ such that $m g, n h \in\mathbb{Z}[x]$, and the gcd’s of the coefficients of $m g, n h$ are 1. Then we have $(m g)(n h) = d f$ for some $d$.