The norm of a Gaussian integer is the natural number defined as N(a + bi) = a^{2} + b^{2}. The norm is multiplicative, i.e. N(zw) = N(z)N(w). The units of Z[i] are therefore precisely those elements with norm 1, i.e. the elements 1, 1, i and i.
The prime elements of Z[i] are also known as Gaussian primes. Some prime numbers are not Gaussian primes; for example 2=(1+i)(1i) and 5=(2+i)(2i). Those prime numbers which are congruent to 3 mod 4 are Gaussian primes; those which are congruent to 1 mod 4 are not. This is because primes of the form 4k+1 can always be written as the sum of two squares, so we have p = a^{2} + b^{2} = (a + bi)(a  bi). If the norm of a Gaussian integer z is a prime number, then z must be a Gaussian prime, since every nontrivial factorization of z would yield a nontrivial factorization of the norm. So for example 2 + 3i is a Gaussian prime since its norm is 4 + 9 = 13.
The ring of Gaussian integers is the integral closure of Z in the field Q(i) consisting of the complex numbers whose real and imaginary part are both rational.
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