I= GROUP ACTIONS ON FIELDS
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Title of Thesis
GROUP ACTIONS ON FIELDS

Author(s)
Muhammad Aslam Malik
Institute/University/Department Details
Department of Mathematics/ University of the Punjab
Session
2002
Subject
Mathematics
Number of Pages
170
Keywords (Extracted from title, table of contents and abstract of thesis)
coset diagrams, ambiguous numbers, ambiguous integers, ambiguous units, ambiguous primes, graphical representation, quadratic field

Abstract
Coset Diagrams are a graphical representation of the permutation action of the groups. Studying groups through their actions on different sets and algebraic structure has become a useful technique to know about the structure of the groups. The graphs have played a vital role in studying these actions.

The main object of this work is to examine the actions of infinite groups G = < x,y; x2 = y3 =1>, H = < x, y; x2 = y4 = 1> and M = < x, y; x2 = y6 = 1> on real quadratic fields Q(ˆšn) and to find the subsets of Q(ˆšn) invariant under the action of each of these groups. Certain proper subsets of Q(ˆšn) invariant under the actions of each of these groups G, H and M, Have also been discussed in this dissertation.

In this dissertation, a type of graphs, called coset diagrams, is employed to investigate the orbits of certain subsets Q*(ˆšp), Q//(ˆšp) and Q///(ˆšp) of Q(ˆšp), p a rational prime, under the actions of the groups G, H and M respectively.

This dissertation is concerned with the determination of number of ambiguous numbers, ambiguous integers, ambiguous units and ambiguous primes in certain subsets Q*(ˆšn), Q//(ˆšn) and Q///(ˆšn) of Q(ˆšn) which are invariant under the action of the groups G, H an M respectively. One of the principal results of this dissertation is that we have determined, for each non square positive rational integer n, the actual number of ambiguous numbers in Q*/(ˆšn), as a function of n.

Download Full Thesis
1822.35 KB
S. No. Chapter Title of the Chapters Page Size (KB)
1 00 Contents
KB
2 0 Introduction And General Terminology 1
93.82 KB
  0.1 Introduction 1
  0.2 General Terminology 3
3 1 Some Basic Definitions And Known Results 6
73.81 KB
  1.0 Introduction 6
  1.1 Quadratic Fields 6
  1.2 Some Results From The Theory Of Numbers 10
  1.3 Graph Theory And Group Actions 14
  1.4 Real Quadratic Irrational Numbers 16
4 2 Groups With Graphical Representations 16
123.97 KB
  2.1 Introduction 17
  2.2 An Account Of Coset Diagrams 17
  2.3 Coset Diagrams And Modular Group G 19
  2.4 Coset Diagrams And The Group H= 22
  2.5 Coset Diagrams And The Group M= 24
5 3 Real Quadratic Fields Under The Action Of The Modular Group G 26
120.13 KB
  3.1 Introduction 26
  3.2 Properties Of Real Quadratic Irrational Numbers Under The Action Of The Group G 26
  3.3 Invariant Subsets Of Q(ˆšN) Under The Action Of The Group G 34
  3.4 The Orbits Of Q*(ˆšP), P A Rational Prime, Under The Action Of The Modular Group G 45
6 4 Real Quadratic Fields Under The Action Of The Group H= 57
329.62 KB
  4.1 Introduction 57
  4.2 Properties Of Real Quadratic Irrational Numbers Under The Action Of The Group H 59
  4.3 Invariant Subsets Of Q(ˆšN) Under The Action Of The Group H 70
  4.4 The Orbits Of Q // (ˆšP), P A Rational Prime, Under The Action Of The Group H 76
7 5 Real Quadratic Fields Under The Action Of The Group M= 83
270.74 KB
  5.1 Introduction 83
  5.2 Properties Of Real Quadratic Irrational Numbers Under The Action Of The Group M 85
  5.3 Invariant Subsets Of Q(ˆšN) Under The Action Of The Group M 95
  5.4 The Orbits Of Q // (ˆšP), P A Rational Prime, Under The Action Of The Group M 103
8 6 Ambiguous Numbers, Integers, Units And Primes Of Certain Subsets Q(ˆšN) 111
309.21 KB
  6.1 Introduction 111
  6.2 Ambiguous Numbers In Q*(ˆšN) 112
  6.3 Integers, Units And Primes In Q*(ˆšN) 124
  6.4 Ambiguous Numbers In Q // (ˆšN), Q**(ˆšN) 126
  6.5 Integers In Q // (ˆšN), Q**(ˆšN) 130
  6.6 Units And Primes In Q // (ˆšN), Q**(ˆšN) 135
  6.7 Ambiguous Numbers In Q /// (ˆšN), Q***(ˆšN) 138
  6.8 Integers In Q /// (ˆšN), Q***(ˆšN) 141
  6.9 Units And Primes In Q /// (ˆšN), Q***(ˆšN) 143
9 7 A Sketch For Further Research 148
485.01 KB
10 8 Bibliography 149
485.01 KB
11 9 Index 153
485.01 KB