A polynomial $F(x)=a_n{x}^n+a_{n‐1}{x}^{n‐1}+\dots +a_{\text{1}}x+a_{\text{0}}$ with integer coefficients is irreducible over the rationals if a prime number p exists such that:

1. (i) p divides each a_i for i ≠ n;

2. (ii) p does not divide a_n;

3. (iii) p² does not divide a₀.

For example, xⁿ − 2 is irreducible for all values of n. The criterion can also be usefully applied to cyclotomic polynomials. It cannot immediately be applied to x² + x + 1, but if we set x = u + 1, then we obtain u² + 3u + 3; this second polynomial is irreducible by the criterion (with p = 3) and, as it is irreducible, so is x² + x + 1.