The Gaussian Integers I: the Fundamental Theorem

Earlier we justified our proof of the fundamental theorem of arithmetic by arguing that the result was not obvious, since its analogue in the ring E of even integers did not hold. Here we will explore a far more exciting explanation: essentially the same proof can be used to show that certain other “integer-like rings” have the unique factorization property. The unique factorization property can be formulated in any integral domain: that is, one has the notion of a unique factorization domain (or UFD). Rather than begin with this general (purely algebraic) notion here, we choose to keep a more properly number-theoretic focus and start with a particular ring which will be very important for us in the rest of the course. Namely, we consider the ring Z[i] of Gaussian integers. Here i = √−1, and the easiest way to view Z[i] is as the subring of complex numbers {a+bi | a, b ∈ Z}, i.e., those complex numbers whose x and y coordinates are integers. In other words, the Gaussian integers are just the lattice points in the plane, added and multiplied as complex numbers. Let us check that this is a subring of C. It contains 0 = 0 + 0i and 1 = 1 + 0i. Moreover it is certainly closed under addition: (a+bi)+(c+di) = (a+c)+(b+d)i, and if all of a, b, c, d are integers, so are a + c and b + d. Finally we have to check closure under multiplication: