UFD to finish the proof.
I had a different proof (NonCommAlg beat me ) that did not involve finding factorizations.
Note that . Therefore, if (a non-unit) is decomposed into irreducibles it must mean that norm of each irreducible is a divisor of . Since is not a prime and it means all irreducibles must have norm . Note that is the only irreducible of norm up to associates. This immediately forces to be a unit for otherwise it would mean are not relatively prime. Thus, for some unit .