Fundamental Theorem of Arithmetic in the Gaussian Integers Malors Emilio Espinosa Lara January 10, 2025 University of Toronto The Gaussian Integers Definition Definition A Gaussian Integer is a complex number a+ bi with a, b ∈ Z. 1 They form a grid We can visualize them as forming a grid in the complex plane. 2 What made all work in Z? What made all our work to function so far? That we were able to divide one integer by another in a way that the residue strictly decreased. 3 What made all work in Z? Definition Let z and w ̸= 0 be Gaussian Integers. We say z is divisible by w if there exists a Gaussian Integer v such that z = wv . 4 What made all work in Z? Example Let z = 3 + 7i and w = −1 + 2i . Then 3 + 7i −1 + 2i = (3 + 7i)(−1− 2i) (−1 + 2i)(−1− 2i) = 11− 13i 5 = 11 5 − 13 5 i . This last number is not a Gaussian Integer. So −1 + 2i does not divide 3 + 7i . 5 What made all work in Z? What made all our work to function so far? That we were able to divide one integer by another in a way that the residue strictly decreased. In the complex numbers we cannot decide consistently who is bigger or smaller. We try to fix this by working with the square of its distance to the origin (because it is always a nonnegative integer!). 6 The norm and the circles We define the norm of a Gaussian Integer as N(a+ bi) = a2 + b2. It is the square of the usual complex norm N(z) = |z |2. 7 Multiplicativity Proposition The norm is multiplicative, that is, N((a+ bi)(x + yi)) = N(a+ bi)N(x + yi). Proof. Call z = x + yi . Then notice that x2 + y2 = (x + yi)(x − yi) = zz = |z |2. Then N(zw) = |zw |2 = (|z ||w |)2 = |z |2|w |2 = N(z)N(w). 8 How do we use the norm? The norm connects the multiplicative structure of the Gaussian Integers with that of the regular integers. It allows us to translate problems in Gaussian Integers into problems of regular Integers. 9 How do we use the norm? Proposition The units of the Gaussian Integers are 1,−1, i ,−i Proof. Suppose u is a unit in the Gaussian Integers.. By definition it means there exists v , in the Gaussian Integers, such that uv = 1. Now take the norm! We get N(u)N(v) = N(1) = 1. Hence, units must have norm 1! There are only four Gaussian integers of norm 1 and by inspection all of them are units. 10 How do we use the norm? This means that every Gaussian Integer z has associated to it 4 trivial multiples z ,−z , iz ,−iz . This is the analogous of a number n having the trivial multiples n and −n. 11 The grid of a number The multiples form a grid of squares along the directions of the lines through 0 and z , and the one 0 and iz . 12 The grid of a number The fundamental square has side length |z |. 13 The grid of a number Any Gaussian Integer is located in one of the squares (possibly in the boundary). 14 The Geometric Fact For every point in a square there is one corner at distance strictly smaller than the sidelength 15 Division with remainder Theorem Let w , z be Gaussian Integers with z ̸= 0. Then there are Gaussian integers q, r such that w = zq + r , and with 0 ≤ N(r) < N(z). Proof. 1. Locate w in the appropriate square of the grid of multiples of z . 2. zq is one of the closest corners. 3. r is the vector from this corner to w . It is a Gaussian Integer. 4. Its norm decreases by the above property of the square. 16 Division with remainder Example Consider z = 3 + 2i and w = 7− 6i . The norms of these Gaussian Integers are N(3 + 2i) = 32 + 22 = 13, N(−7− 6i) = (−7)2 + (−6)2 = 85. Thus 7− 6i is the large one. So we divide by 3 + 2i . 17 Division with remainder Example We create the grid of multiples of 3 + 2i . We notice that −3(3 + 2i) is the closest multiple to −7− 6i . 18 Division with remainder Example The difference is the residue. We have r = −7− 6i − (−3(3 + 2i)) = −7− 6i + 9 + 6i = 2. Thus −7− 6i = −3(3 + 2i) + 2,N(2) = 4 < 13 = N(3 + 2i). 19 Fundamental Theorem of Arithmetic We can now proceed in exactly the same way. Definition A Gaussian Integer is composite if it can be written as a product of two non-unit Gaussian Integers. A Gaussian Integer is prime if is not composite and is not a unit. 20 Fundamental Theorem of Arithmetic We now conclude, by the same process as for Z: Theorem Every Gaussian Integer that is not a unit has a unique prime factorization up to units.
