Z[sqrt(-3)] does not have unique factorization. However, Z[w] does, where w = (-1 + sqrt(-3))/2 denotes a primitive third root of unity. In particular, 2

Z[sqrt(-3)] does not have unique factorization. However, Z[w] does, where w = (-1 + sqrt(-3))/2 denotes a primitive third root of unity. In particular, 2 * 2 = (-1 + sqrt(-3)) * (-1 - sqrt(-3)) gives two irreducible factorizations of 4 in Z[sqrt(-3)] (you can use norm arguments in Z[w] to show irreducibility). Note that the right hand side can also be written as (2w) * (2w^2) however, since w is not in Z[sqrt(-3)], the two factorizations above are distinct in Z[sqrt(-3)]. They become the same factorization in Z[w]. Edit: Another way to think about why Z[sqrt(-3)] doesn't have unique factorization is because it isn't integrally closed[1]--that is, there is a monic polynomial (ie a polynomial with a leading coefficient of 1) with coefficients in Z[sqrt(-3)] that doesn't have roots in Z[sqrt(-3)]. In particular, since w^2 + w + 1 == 0, w is a root of the polynomial x^2 + x + 1, which is monic over Z[sqrt(-3)]. It turns out that any unique factorization domain[2] is integrally closed. Since Z[sqrt(-3)] is not integrally closed, it is not a UFD. [1]: https://en.wikipedia.org/wiki/Integrally_closed_domain https://en.wikipedia.org/wiki/Integrally_closed_domain [2]: https://en.wikipedia.org/wiki/Unique_factorization_domain https://en.wikipedia.org/wiki/Unique_factorization_domain