A Particular Euclidean Ring

Let



We call these the Gaussian integers. Our first objective is to exhibit as a Euclidean ring. In order to do this we must first introduce a function defined for every nonzero element in which satisfies

1.  is a nonnegative integer for every

2.  for every in

3. Given there exist such that where or

Put





LEMMA 3.8.1: Let be a prime integer and suppose that for some integer relatively prime to we can find integers and such that



Then
can be written as the sum of squares of two integers, that is, there exist integers and such that




LEMMA 3.8.2: If is a prime number of the form then we can solve the congruence



THEOREM 3.8.2 (Fermat): If is a prime number of the form then



for some integers