TITLE: Prove that the polynomial $\prod\limits_{i=1}^n\,\left(x-a_i\right)-1$ is irreducible in $\mathbb{Z}[x]$.
QUESTION [6 upvotes]: Let $n>1$ be an integer.  For $a_1,a_2,\ldots,a_n\in\mathbb{Z}$ with $a_1< a_2< a_3 < \dots < a_n$, prove that the polynomial $$f(x)=(x-a_1)(x-a_2)\cdots(x-a_n)-1\,.$$ is irreducible in $\mathbb{Z}[x]$.

Please help! Thanks!

REPLY [2 votes]: This is an alternative answer to your question.
Consider the polynomial $f^2$ instead. So $f^2|_{\{a_i\}}=1$. If $f$ is reducible over $\mathbb{Z}[x]$, then it must be reducible to $f^2=g^2h^2$ for $g,h \in \mathbb{Z}[x]$. So $f^2=g'h'$. Suppose $\deg(g')\leq n$. $g'|_{\{a_i\}}=1$ as $g$ is positive and $g'(a_i)$'s are all invertibles in $\mathbb{Z}$. So $(g'-1)|_{\{a_i\}}=0$ implies $a_i$ are all its roots. Since $\deg(g)\leq n$, $g'=\prod_i(x-a_i)+1$ by Bézout's theorem. Similarly for $h'$. Now $h'g'\neq f^2$. Thus $f$ is irreducible over $\mathbb{Z}[x]$.