## 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 $$d> 1$$, 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 $$d$$.

Ben Lynn blynn@cs.stanford.edu 💡