Hi I am very new to doing math proofs so I need some help to tell me if what I did is write or not. Here is the question and my solution for it but is it the right way to do it? If not, what am I missing? What did I do wrong? Thank you for the help.
Question : Use Gauss's Lemma to show that if $f\in\ \mathbb{Z}[x]$ is irreducible, then it remains irreducible when viewed as an element of $\mathbb{Q}[x]$.
Let $f_z \in \mathbb{Z}[x]$ be irreducible(and hence primitive) and $f_q \in \mathbb{Q}[x]$ be reducible and $f_z=af_q$ for some $a\in \mathbb{Z}$. Since $f_q$ is reducible we have $f_1(x), f_2(x) \in \mathbb{Q}[x]$ such that $f_q=(f_1f_2)$ and $f_1,f_2$ are not invertible. But $$f_z=af_q=af_1f_2$$ which is a contradiction as $f_z$ is primitve and by Gauss's Lemma $f_1,f_2$ do not exist and hence our assumption is false.