Theorem 1 (Eisenstein criterion)   Let $f$ be a primitive polynomial over a commutative unique factorization domain $R$ , say $$f(x)=a_0 + a_1x + a_2x^2 + \ldots + a_nx^n \;.$$ If $R$ has an irreducible element $p$ such that $$p\mid a_m \qquad 0\le m\le n-1$$ $$p^2 \nmid a_0$$ $$p \nmid a_n$$ then $f$ is irreducible.

Proof. Suppose $$f=(b_0 + \ldots + b_s x^s)(c_0 + \ldots + c_t x^t)$$ where $s>0$ and $t>0$ . Since $a_0 = b_0 c_0$ , we know that $p$ divides one but not both of $b_0$ and $c_0$ ; suppose $p \mid c_0$ . By hypothesis, not all the $c_m$ are divisible by $p$ ; let $k$ be the smallest index such that $p\nmid c_k$ . We have $a_k = b_0 c_k + b_1 c_{k-1} + \ldots + b_k c_0$ . We also have $p\mid a_k$ , and $p$ divides every summand except one on the right side, which yields a contradiction. QED