Perron's irreducibility criterion

Remarks

Proof idea: by Rouché's theorem, comparing $f$ with $a_{n-1} x^{n-1}$ on the unit circle, $f$ has exactly $n - 1$ roots in the open unit disc and one root outside it. If $f = g h$ with monic $g, h \in \Z[x]$ of positive degree, one factor has all its roots in the open unit disc. Its constant term is then a nonzero integer of absolute value less than $1$, which is impossible. Irreducibility over $\Q$ follows by Gauss's lemma. Example: $x^3 + 5x^2 + 2x + 1$ is irreducible, since $5 > 1 + 2 + 1$. The hypothesis $a_0 \ne 0$ is necessary: $x^2 + 3x = x(x + 3)$ satisfies the inequality $3 > 1 + 0$. A second form of the criterion allows equality in the inequality, provided that $f(1) \ne 0$ and $f(-1) \ne 0$.

Going deeper

For a student

The criterion is a typical analytic proof of irreducibility. One locates the complex roots, then uses that the constant term of an integer factor is a nonzero integer. It complements algebraic tests such as Eisenstein's criterion and reduction modulo a prime. For example, it shows at once that $x^n + 3x^{n-1} + 1$ is irreducible for every $n \ge 2$.