ABSTRACT
When you try to solve a polynomial equation whose coefficients are integers, its solutions are often not integers. In order to solve 2x+ 5 = 0, x2 + 3x− 1 = 0, or x2 + 1 = 0, you have to go beyond the integers and work in the rational, real, or complex numbers. We encounter the same scenario in number theory, where we often find it natural to use irrational or complex numbers, even when trying to find integral solution to equations. In this chapter, we look at numbers of the form a + bi, where i =
√−1, and where a and b are integers. These numbers are called Gaussian integers and are usually denoted Z[i]:
Z[i] = {a+ bi | a, b are integers}
(Z is a standard notation for the integers, from the German word Zahlen, meaning numbers). Gauss used Z[i] when he worked on Biquadratic Reciprocity, his generalization of Quadratic Reciprocity to 4th powers. The Gaussian integers have many properties in common with the integers and can be studied in their own right. However, in the last section of this chapter, we give three applications where they arise naturally when looking at problems involving the usual integers.
