Divisibility in $\mathbb{Z}[X]$

Let $$f,g \in \mathbb{Z}[X]$$ be monic polynomials such that $$\deg f > \deg g$$ and $$g$$ is irreducible in $$\mathbb{Z}[X]$$. If there is $$a \in \mathbb{C}$$ such that $$f(a)=g(a)=0,$$ prove that $$g \mid f$$ in $$\mathbb{Z}[X].$$

Since $$g$$ is irreducible in $$\mathbb{Z}[X],$$ it is also irreducible in $$\mathbb{Q}[X],$$ so $$(f,g)=1$$ or $$(f,g)=g$$ in $$\mathbb{Q}[X].$$
If $$(f,g)=1,$$ since there are $$u,v \in \mathbb{Q}[X]$$ such that $$uf+vg = (f,g)=1,$$ we would get $$0=u(a)f(a)+v(a)g(a)=1$$ which is absurd. Hence $$(f,g) = g$$ in $$\mathbb{Q}[X] \Rightarrow g \mid f$$ in $$\mathbb{Q}[X].$$

How can I extend the divisibility to $$\mathbb{Z}[X]$$? I have huge problems when dealing with $$\mathbb{Z}[X],$$ since $$\mathbb{Z}$$ is not a field, only a ring. Therefore properties like $$uf+vg=(f,g)$$ or $$f=gc+r$$ are not valid.
The moment I see $$\mathbb{Z}[X]$$ I panic and don't know how to approach problems like this one, especially when it comes to divisibility in $$\mathbb{Z}[X].$$ What are the properties from $$\mathbb{F}[X]$$, where $$\mathbb{F}$$ is a field, that work in $$\mathbb{Z}[X]$$ as well?

• This might be relevant. Do you know Gauss' lemma for polynomials? – Crostul Apr 8 at 17:46
• Yes, I know it! Thank you very much! – AndrewC Apr 8 at 17:58