## Electronic Theses and Dissertations

#### Document Type

Thesis - Open Access

1972

#### Degree Name

Master of Science (MS)

#### Department / School

Mathematics and Statistics

#### Abstract

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.

Numbers, Complex
Integrals

application/pdf

37

#### Publisher

South Dakota State University

COinS