Eisenstein's criterion

Eisenstein's criterion: Let $R$ be a UFD and let $p\in R$ be a prime element. Consider $f(x)=a_nx^n+\cdots+a_0\in R[x]$ in the polynomial ring $R[x]$. If

• $p\mid a_i$ for all $i<n$,
• $p\nmid a_n$, and
• $p^2\nmid a_0$,

then $f$ is irreducible in $\mathrm{Frac}(R)[x]$, where $\mathrm{Frac}(R)$ is the fraction field of $R$. Consequently, $f$ is irreducible in $R[x]$.