By definition, N1() = 32 + 22 = 13, and because and are necessarily nonzero (since they are factors of the nonzero number ), their norms are positive integers. Thus we have a factorization 13 = N1()N1() in N. As 13 is a prime number in the usual sense, we must have N1() = 1 or N1() = 1; in the first case, is a unit in Z[i], and in the second case, is a unit in Z[i]. Having proved that the hypothetical factorization of in Z[i] is trivial, we may conclude that is prime. 1. Assume that d Z and d / Q. Prove that if Z[ d] has the property that |Nd()| is a prime number, then is a prime of Z[ d]. (Hint: take inspiration from the example above.)