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.
If is a field extension of and is algebraic over (i.e., the smallest subextension containing , , is finite-dimensional over ), then the -algebra map has a non-trivial kernel which is a maximal ideal, which being principal is generated by an irreducible polynomial . The unique monic polynomial generator is called the irreducible polynomial of .