The Gaussian Integers

Since the work of Gauss, number theorists have been interested in analogues of Z where concepts from arithmetic can also be developed. The example we will look at in this handout is the Gaussian integers: Z[i] = {a + bi : a, b ∈ Z}. Excluding the last two sections of the handout, the topics we will study are extensions of common properties of the integers. Here is what we will cover in each section: (1) the norm on Z[i] (2) divisibility in Z[i] (3) the division theorem in Z[i] (4) the Euclidean algorithm Z[i] (5) Bezout's theorem in Z[i] (6) unique factorization in Z[i] (7) modular arithmetic in Z[i] (8) applications of Z[i] to the arithmetic of Z (9) primes in Z[i] 1. The Norm In Z, size is measured by the absolute value. In Z[i], we use the norm. Definition 1.1. For α = a + bi ∈ Z[i], its norm is the product N(α) = αα = (a + bi)(a − bi) = a 2 + b 2. For example, N(2 + 7i) = 2 2 + 7 2 = 53. For m ∈ Z, N(m) = m 2. In particular, N(1) = 1. Thinking about a + bi as a complex number, its norm is the square of its usual absolute value: |a + bi| = a 2 + b 2 , N(a + bi) = a 2 + b 2 = |a + bi| 2. The reason we prefer to deal with norms on Z[i] instead of absolute values on Z[i] is that norms are integers (rather than square roots), and the divisibility properties of norms in Z will provide important information about divisibility properties in Z[i]. This is based on the following algebraic property of the norm. Theorem 1.2. The norm is multiplicative: for α and β in Z[i], N(αβ) = N(α) N(β). Proof. Write α = a + bi and β = c + di. Then αβ = (ac − bd) + (ad + bc)i. We now compute N(α) N(β) and N(αβ): N(α) N(β) = (a 2 + b 2)(c 2 + d 2) = (ac) 2 + (ad) 2 + (bc) 2 + (bd) 2 1