Prove that polynomials $f(x)=\prod_{i=1}^{n} (x-a_i)^{2}+1$ and $g(x)=\prod_{i=1}^{n} (x-a_i)-1$ are irreducible over $\mathbb Q$, where $a_i \in \mathbb Z, a_i \neq a_j$ for $i \neq j$.
I've seen the proof for irreducibility over $\mathbb Z$. But I don't know what to do in this case. For $f(x)$ i got $f=f_1*f_2$, then $f_1(a_i)f_2(a_i)=1$, and from this I got that $f_1+1/f_2$ got $n$ roots over $\mathbb Q(x)$.