Document Type

Thesis - Open Access

Award Date


Degree Name

Master of Science (MS)


Mathematics and Statistics


Throughout this paper, small case Latin letters, with the exception of i which has its usual mathematical meaning, will denote rational integers where we let Z represent the set of all rational integers. We call µ = a + bi a Gaussian integer iff a and b are rational integers and we denote the set of all Gaussian integers by Z(i). The set of Gaussian integers can be represented geometrically by ·the set of lattice points in a Cartesian coordinate system whose horizontal and vertical grid lines are one unit apart. Since µ = a + bi is a complex number, it has a complex conjugate denoted byµ= a - bi. The norm of µ, written as N(µ), is defined as N(µ) = µµ = a2 + b2. It is a trivial matter to show that the noun is multiplicative so that N(aB) = N(a)N(B). We say that a ≠ 0 divides. B iff there exists a µ such that B = aµ, we write a/B. A Gaussian integer w is called a unit iff w/a for all a. Since the norm is multiplicative, it can be shown that w is a unit iff N(w) = 1. Hence, the units of Z(i) are ± 1 and ± i. For any µ, we call wµ an associate iff w is a unit. A Gaussian integer p is said to be prime iff whenever p = aB , one of a or B is a unit, but not both. We call p a real prime iff p is a prime in Z(i) and Z. In contrast, primes in Z which are not primes in Z(i) are called rational primes. It is obvious that all of the associates of p as well as P are prime if P is prime. For any Gaussian integer µ, we say that a and B are congruent modulo µ, written a = B (mod µ), iff µ/ (a - B). Since congruence modulo µ is an equivalence relation on the set Z(i), it partitions Z(i) into a collection of pairwise disjoint sets whose union is Z(i). Hence, as is the case with the rational integers, we define a complete residue system modulo μ as a nonempty collection S of elements of Z(i) such that (1) no two elements of S are congruent modulo µ, and (2) every element of Z(i) not in S is congruent to some element of S. A complete residue system modulo µ is abbreviated as C.R.S.(mod µ). The purpose of this paper is to investigate the distribution of primitive roots in several special fields of order p2.

Library of Congress Subject Headings

Numbers, Complex



Number of Pages



South Dakota State University