A non-zero polynomial with coefficients in a field is irreducible if when written as the product of two polynomials, one of or is a constant (and necessarily non-zero) polynomial. Equivalently, a polynomial is irreducible if the ideal it generates is a maximal ideal of the polynomial ring .
Notice that under this definition, the zero polynomial is not considered to be irreducible. An alternative definition, which applies to the case of coefficients in a commutative ring , is that a polynomial is irreducible if, whenever divides , either divides or divides . Under this definition, a polynomial is irreducible if it generates a prime ideal in , and the zero polynomial is irreducible if is an integral domain.