State and prove Eisenstein’s criterion.
Solution
Let \[ f(x)=a_nx^n+\cdots+a_1x+a_0\in\ZZ[x] \] be primitive. Suppose there is a prime \(p\) such that \[ p\nmid a_n, \qquad p\mid a_i\quad(0\le i<n), \qquad p^2\nmid a_0. \] Then \(f\) is irreducible in \(\QQ[x]\).
By Gauss’s lemma it is enough to prove irreducibility in \(\ZZ[x]\). Suppose \[ f=gh \] with nonconstant \(g,h\in\ZZ[x]\). Reduce modulo \(p\). Since all nonleading coefficients of \(f\) vanish modulo \(p\), \[ \bar f=\bar a_n x^n \] in \(\FF_p[x]\). Thus \[ \bar g=cx^r, \qquad \bar h=dx^s \] for some \(r,s\ge1\) and nonzero \(c,d\in\FF_p\).
Therefore the constant terms of both \(g\) and \(h\) are divisible by \(p\). Their product is \(a_0\), so \[ p^2\mid a_0, \] contradicting the hypothesis.
Hence no nontrivial factorization exists, and \(f\) is irreducible over \(\QQ\).