Let and let be the ring of integers in . Factor the ideals and in .

We claim that . Indeed, , so that . The reverse inclusion is clear.

Now we claim that is maximal. By Corollary 9.11, , and by Theorem 9.14, we have , so that . By Corollary 9.15, is a prime ideal.

Thus is the prime factorization of in .

Now we claim that . Indeed, we have .

Next we claim that and are proper. Suppose to the contrary that ; then we have for some . Comparing coefficients, and , which yields a contradiction mod 5. So is proper. Likewise, is proper. In particular, neither nor have norm 1 as ideals. Now , and neither factor is 1, so that . By Corollary 9.15, and are prime ideals.

Thus is the prime factorization of in .