Eisenstein Irreducibility Criterion

definitionring-theory

Definition

Proof

Consider $\phi :\mathbb{Z}\to {\mathbb{Z}}_p$ defined by $\phi (n)=n\mathrm{mod}p$ $\varphi :\mathbb{Z}[x]\to {\mathbb{Z}}_p[x]$ induced morphism

Assume $P(x)=A(x)B(x)$, may assume $A,B$ monic of positive degree. Then $\varphi (P)=\varphi (A)\cdot \varphi (B)$ in ${\mathbb{Z}}_p[x]$ (field, euclidean domain, principal ideal domain, and unique factorization domain)…

Examples