Units in some algebraic number fields : a thesis presented in partial fulfilment of the requirements for the degree of Master of Science in Mathematics at Massey University.
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Dirichlet's theorem describes the structure of the group of units of the ring of algebraic integers of any algebraic number field. This theorem shows that any unit can be written in terms of a fundamental system of units. However Dirichlet's theorem does not suggest any method by which such a fundamental system of units (or indeed any units) can be obtained. This thesis looks at three types of algebraic number fields for which a fundamental system of units contains one unit, the so called fundamental unit. In each case properties of units and the problem of obtaining a fundamental unit are discussed. Chapter one is an introductory chapter which summarises the basic theory relevant to algebraic number fields of arbitrary degree. Basic properties of units and Dirichlet's theorem are also given. Chapter two looks at units of Quadratic fields, Q(√d). Units of imaginary quadratic fields are mentioned briefly but the chapter is mainly concerned with the more complicated problem of obtaining real quadratic units. The relevant theory of simple continued fractions is presented and the way in which units can be obtained from the simple continued fraction expansion of √d is outlined. The chapter then also looks at some recent papers dealing with the length of the period of √d and concludes by showing how units can be obtained from the simple continued fraction expansion of (1 + √d)/2 when d ≡ 1(mod 4). Chapter three looks at units of pure cubic fields. The basic properties of pure cubic units are developed and reference is made to various algorithms which can be used to obtain pure cubic units. The main purpose of this chpater is to present the results of the paper 'Determining the Fundamental Unit of a Pure Cubic Field Given any Unit' (Jeans and Hendy [l978]). However in this thesis a different approach to that of the paper is used and for two of the results sharper bounds have been obtained. Several examples are given using the algorithm which is developed from these results. Chapter four, which is original work, investigates the quartic fields, Q(d¼), where d is a square-free negative integer. Similarities between these quartic fields and the pure cubic and real quadratic fields are developed of which the main one is a quartic analogue of the results given in the paper mentioned above. The examples given in chapter three required multiprecision computer programs and these programs have been listed in appendix one.