Eisenstein criterion

Proving a polynomial is irreducible is usually hard — you must rule out every possible factorization. Eisenstein's criterion is a beautifully cheap shortcut: glance at the coefficients, check a single prime's divisibility pattern, and if it matches, irreducibility is guaranteed instantly.

Let f(x) = a_n x^n + ... + a_1 x + a_0 have coefficients in an integral domain R, and let p be a prime ideal of R. Suppose p does not contain the leading coefficient a_n, p contains every other coefficient a_0, ..., a_{n−1}, and p^2 (the ideal of products of two elements of p) does not contain the constant term a_0. Then f is irreducible over the field of fractions of R. Over the integers this reads: a prime p divides all coefficients except the leading one, and p^2 does not divide the constant term.

The criterion is sufficient but not necessary — many irreducible polynomials fail it. Its power is amplified by combining it with substitutions: if f(x) is not obviously Eisenstein, f(x + c) might be. The standard triumph is the p-th cyclotomic polynomial 1 + x + ... + x^{p−1}, which becomes Eisenstein at p after substituting x ↦ x + 1, proving it irreducible over Q.