ABSTRACT
In this chapter we will investigate O(√−1), known as the Gaussian integers . We recall that
O(√−1) = {a+ b√−1 | a and b are integers} = {a+ bi | a and b are integers},
and we have shown that O(√−1) is a UFD. We will begin by proving a justly famous theorem of Fermat: every prime congruent to 1 (mod 4) can be written in as a sum of squares of two positive integers, p = x2 + y2, uniquely up to the order of the summands. (For example, 5 = 22 + 12, 13 = 32 + 22, 17 = 42 + 12, 29 = 52 + 22, 37 = 62 + 12, 41 = 52 + 42). We shall present three proofs of this theorem in the first section of this chapter.
