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

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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

Neville Stuart Jeans

19 84

in Mathematics at Massey University

ii

ABS T R ACT

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(o). Our attention is mainly centered on type I quartic fields, that is quartic fields for which Q(o) 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 ide� 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(o) and show many similarities to the simple continued fraction algorithm. (Existing work of this nature restricts its attention to cases where Q(o) has class number one).

We obtain an algorithm which is periodic for complex numbers w satisfying

iii w 2 E Q (o) , w � Q (a) . This enables us to calculate t.mits of type I quartic fields .

In chapter three we consider quartic fields Q (r) which are a quadratic extension of a quadratic field Q (o). In section one we express the ring of integers of Q (r) in tenns of the integers of Q (o) 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 (r) 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(r) is of the second simplest form . For the two remaining fonns we obtain a unit U which may or may not be

fundamental . We therefore develop an algorithm which calculates a fundamental tmi t from U . To illustrate the use of our algorithms we calculate fundamental tmits for the type I quartic fie lds

Q (�) , D E Z, -99

�

D

^�

-1

Finally in section three we consider the calculation of a ftmdamental

system of units for type I Ib 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 .

ACKNOW LEDG EMENT S

I would like to thank my supervisors Dr M . D . Hendy and Dr

K.L.

^{T eo}

for the advice and encouragement they have offered during the preparation of this thesis. Thanks also to Gail Tyson for typing this thesis.

CONTENTS

REFERENCE PAGES FOR NaTATION

^AND

TERMINOLOGY rnAPTER ONE

Section One Section Two

ALGEBRAIC NUMBER FIELDS, QUADRATIC FIELDS,

AND

SIMPLE CONTINUED FRACTIONS

Algebraic Number Fields , Modules , and Miscellanea

Quadratic Fields

Section Three Simple Continued Fractions and Real Quadratic Units

Section Four Relative Minima of Z Modules

OiAPTER TWO RELATIVE MINIMA OF MODULES OVER A RING OF CQ\1PLEX QUADRATIC INTEGERS

Section One Definitions , Notation, and Basic Theorems Section Two Order of Approximation of a Relative Minimum Section Three

^An

Algorithm for the Calculation of

Relative Minima

Section Four Periodic Relative Minima Section Five Comparisons and Conclusions

rnAPTER TIIREE UNITS OF CERTAIN QUARTIC EXTENSIONS OF Q HAVING A QUADRATIC SUBFIELD

Section One Quartic Fields having a Quadratic Subfield, their Integers and a Classification

Section Two Units of Type I Quartic Fields Section Three Units of Type IIb Quartic Fields REFERENCES

vi

1

1 6

12 20

2 5

2 5 4 2

4 8 87 123

133

133 156 19 3

2 0 7

vi

R EF ER ENCE P AG ES F OR NOT AT I ON AND T ER MINOLOGY

The list which follows is not exhaustive. We have omitted most standard notation and terminology plus most notation and terminology which has a localised usage.

Q(a) , Z (a) , Q(a) (S) , Z (a) (S) . 3 [Q(a.) :Q], N , N a . . . . 3

R[A1, . . . ,'\] where R=Z , Z (a) . 5

packing constant 5

d, o, Q(o) , Z (o) . 6

c . 6

w

a'

E(d)

<

al' az' . . .

'ak>

I

^{I • •}

CMp) h(d)

{x}

z (o) +

A*(reverse) , !AI ,A .

M(w) ,

^{W • •}

M(w)/X, (M(w)/X)/Y relative minimum

Aa ... .

. 7 . 7 7 8

8 8 8 9 1 1 1 2 2 6 26 2 7

2 8 3 1

�,

equivalent relative minima 3 2 Ek, J\ . . . . . . . . 3 2

A � j) ' �' �' � j )' � j) 3 3 chain, complete chain, half

chain 33

�(w) , �,j , Rk,j , \ 49

basic, non-basic 5 2

Ik, gk' crk

⁼

(ak+o)/c 5 2 Kk, Ak, Bk, wk . . . . 5 3

standard representation of � (w) 5 3

(a,S) Wk allowable 5 3

lj!k

wk

'

1

^'

wk

' '

2 nk ' 1 ' n. .K' 2

nk

'

3' nk ' 4' � · · · ·

5 3 5 7 66

periodic relative minima, period, length, q

N0(R) for R E M(w)/A ek . . .

tmit of M(w) minimal period M*(w) , Mk (w)

symmetric minimal period

z

c o)

^CF

Q(r)

non-square and rational square-free

form 1 , 2 , 3 , 4

89 9 1 9 3 9 4

96 99 99 1 0 1 . . 125 133

. . 133 . . 134

. . 150

type I , II quartic fields . . . 1 5 1

type I I a , l ib , I I c quartic

fie lds . 1 53

1 5 7 160

Tf

⁼

(lll+llzwm) /�, b(llz)

^•

L ( IY) • •

D

⁼

rs 2t3

A*, A' ,A'*, A*'

vii

16 2

. 1 74

183

. 193