Irreducible polynomials are used to generate new fields using the ring of cosets formed by the ideal generated by the irreducible polynomial. Polynomials of degree two or three are irreducible if and only if they have no linear factors and hence no roots. Polynomials of degree four are thus the first interesting case. If there is a prime such that the polynomial is irreducible in the finite field , then is irreducible over the integers.

However, there are irreducible polynomials over the integers that factor over all primes. The Gaussian integers are an alternative number system used to check for factorization.