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 $R[x]$.

Proof sketch: If $f=gh$ with positive degrees, reduce coefficients modulo $p$. The hypotheses force $\bar f(x)=\bar a_n x^n$ in $(R/(p))[x]$, so one factor has zero constant term mod $p$. Tracking constant terms then forces $p^2\mid a_0$, contradicting the assumption.