Prove that every prime integer congruent to 1 mod 4 is a sum of two squares. Show that every product of two such primes is also a sum of two squares.
Let be such a prime. We know that is not irreducible in ; thus there exist nonunits such that . Note that neither of and can have norm 1. Since , letting , we see that as desired.
Now suppose are integer primes congruent to 1 mod 4. As above, we have and , where and have norm and as needed. Now . In particular, if , then .