Irreducible polynomial.

Question

Prove that polynomials $p(x)=\left((\prod_{i=1}^{n} (x-a_i))^{2}\right)+1$ and $q(x)=\left(\prod_{i=1}^{n} (x-a_i)\right)-1$ are irreducible over $z$ ,where $a_i $ ' s are distinct integers .

Answer 1

Hint for the irreducibility of $q$.

Assume that $q=fg$, where $f,g\in\mathbb{Z}[x]$, they are monic and of degree $<n$. Then for $i=1,\dots,n$, $$q(a_i)=f(a_i)g(a_i)=-1$$ which implies that the integers $f(a_i)$ and $g(a_i)$ have to be equal to $\pm 1$, and $f(a_i)=-g(a_i)$.

Now consider the polynomial $f+g$ and try to get a contradiction.

Answer 2

For $p(x)$, Let if possible $p(x)=q_1 (x)q_2 (x) $ and $ p(a_i)=1$ for n distinct integers. So each $q (a_i)$ is equal to 1,due to fact that $p (x) \geq 1$ , so each $q $ has degree $n $. Let $h (x)=q_1(x)-q_2 (x) $and $h (x) $has n distinct zeros , so degree of $h $ is atleast $n $, And leading coefficient of $q_1$ and $q_2$ are equal to 1,this implies $q_1 -q_2$ has degree atmost $n-1$, which is contradiction.