In mathematics, an irreducible polynomial
is, roughly speaking, a non-constant polynomial that cannot be factored into the product of two non-constant polynomials. The property of irreducibility depends on the nature of the coefficients that are accepted for the possible factors, that is, the field or ring to which the coefficients of the polynomial and its possible factors are supposed to belong. For example, the polynomial x2 − 2
is a polynomial with integer coefficients, but, as every integer is also a real number, it is also a polynomial with real coefficients. It is irreducible if it is considered as a polynomial with integer coefficients, but it factors as
if it is considered as a polynomial with real coefficients. One says that the polynomial x2 − 2
is irreducible over the integers but not over the reals.