Calculation of fundamental units in some types of quartic number fields : a thesis presented in partial fulfilment of the requirements for the degree of Doctor of Philosophy in Mathematics at Massey University

Abstract

Dirichlet's theorem describing the structure of the unit group of the ring of integers of an algebraic number field shows that the units are generated by a primitive root of unity of the field plus a finite set of units called a fundamental system of units. However Dirichlet's theorem does not suggest any method by which a fundamental system of units can be obtained. In this thesis we consider the problem of calculating a fundamental system of units for certain types of quartic field which are a quadratic extension of a quadratic field Q(δ). Our attention is mainly centered on type I quartic fields, that is quartic fields for which Q(δ) is complex. In such cases a fundamental system of units contains a single unit called a fundamental unit. To calculate fundamental units of type I quartic fields we use the simple continued fraction algorithm, real quadratic field case as a guide. This topic is reviewed in chapter one where we also note Voronoi's view of simple continued fractions in terms of relative minima of a Z module. In chapter two we consider the idea of relative minima of a module over a ring of complex quadratic integers. Basically we generalize the simple continued fraction algorithm which calculates best approximations to a real number using rational integer coefficients to an algorithm which calculates best approximations to a complex number using complex quadratic integer coefficients. The ideas are developed with respect to an arbitrary complex quadratic field Q(δ) and show many similarities to the simple continued fraction algorithm. (Existing work of this nature restricts its attention to cases where Q(δ) has class number one). We obtain an algorithm which is periodic for complex numbers w satisfying w2 є Q(δ), w ∉ Q(δ). This enables us to calculate units of type I quartic fields. In chapter three we consider quartic fields Q(Γ) which are a quadratic extension of a quadratic field Q(δ). In section one we express the ring of integers of Q(Γ) in terms of the integers of Q(δ) thereby recognising four forms which these rings may take. In section two we consider the problem of calculating fundamental units of type I quartic fields. The algorithm developed in chapter two is only guaranteed to locate a fundamental unit when the ring of integers of Q(Γ) is of the simplest of the four forms mentioned above. A modified version of the algorithm allows us to calculate a fundamental unit when the ring of integers of Q(Γ) is of the second simplest form. For the two remaining forms we obtain a unit U which may or may not be fundamental. We therefore develop an algorithm which calculates a fundamental unit from U. To illustrate the use of our algorithms we calculate fundamental units for the type I quartic fields Q(4√D), D Є Z, -99 ≤ D ≤ -1 Finally in section three we consider the calculation of a fundamental system of units for type IIb quartic fields, that is semi-real quartic fields which are a quadratic extension of a real quadratic field. A connection between type IIb and type I quartic fields enables us to calculate fundamental systems of units for type IIb quartic fields.