adj. 1. (of a polynomial over a field) unable to be factorized into polynomials of lower degree over the same field; for example, (x² + 1) is irreducible over the reals, but can be factorized as (x + i)(x - i) over the complex numbers. Such a polynomial is also prime. 2. (of a non-zero element in an integral domain) not expressible as a product of two non-units; if a = bc then either b or c is a unit. All other non-zero elements are either units or are reducible. The number 5 is irreducible in the integers but since 5 = (2 + i)(2 - i), it is reducible as a Gaussian integer. By a theorem of Fermat, this is also true of any prime that is congruent to 1 modulo 4, being expressible as a sum of two integral squares. Compare prime. 3. (of a radical) unable to be expressed as a rational expression; for example, √(x + 1) is irreducible.