abstract algebra - gcd of two monic polynomials

I am trying to answer the following question but I'm having some trouble:

If $f$ and $g$ are monic polynomials in $Z[x]$, does their (monic) greatest common
divisor in $Q[x]$ necessarily have coefficients in $Z$?

I've tried using Gauss's lemma to get an answer but haven't really gotten anywhere. Any help would be appreciated, Thanks!

Answer

First, we show that a monic factor of a monic polynomial is always in $\mathbb Z$

Suppose, $f(x)=g(x)\cdot h(x)$

Choose $k,l\in \mathbb Z$, such that $g'(x):=k\cdot g(x)$ and $h'(x):=l\cdot h(x)$ are primitive. Then, we have $kl\cdot f(x)=g'(x)\cdot h'(x)$

Because of Gauss's Lemma, $kl\cdot f(x)$ is primitive. But $kl$ is a common factor of the coefficients of $kl\cdot f(x)$. Hence, $kl$ must be a unit and therefore $g(x)$ and $h(x)$ must be in $\mathbb Z[X]$

This implies that a monic greatest common divisor of monic polynomials in $\mathbb Z[X]$ must be in $\mathbb Z[X]$.