APPLICATION OF EULER’S PHI-FUNCTION IN ABSTRACT ALGEBRA BY PATIENCE AHIAKU (BSc. Mathematics) A Thesis Submitted to the department of Mathematics, Kwame Nkrumah University of Science and Technology in Partial Fulfillment of the Degree of MASTER OF PHILOSOPHY (PURE MATHEMATICS) College of Science April, 2014 DECLARATION I hereby declare that this submission is my own work towards the MPhil and that, to the best of my knowledge, it contains no material previously published by another person nor material, which has been accepted for the award of any degree of the university, except where due acknowledgement has been made in the text. Patience Ahiaku (PG5069510) ----------------------- ---------------------- Signature Date Certified by: Dr. S. A. Opoku ----------------------- ---------------------- Supervisor Signature Date Certified by: Prof. Samuel K. Amponsah ----------------------- ---------------------- Head of Department Signature Date ii DEDICATION I dedicate this thesis to the Almighty God and my mother Mrs Matilda Ahiaku iii ACKNOWLEDEMENT My deepest gratitude goes to the Almighty God for seeing me through this work successfully. I could not but say, God bless my Supervisor Dr. S. A. Opoku for his supervision throughout the study. Also I am grateful to Mrs Mary Osei Fokuo Boakye and Dr. R.Y. Tamakloe of Department of Physics for their support and encouragement. My heartfelt appreciation goes to my family for their support and prayers. Finally, to all, who have contributed in diverse ways in ensuring the successful completion of this work. I say God richly bless you all. iv ABSTRACT Euler’s phi – function ( ) is defined for every positive integer as follows: ( ) and when then ( ) is the number of distinct integers such that and are relatively prime. This thesis seeks to examine the application of Euler’s phi-function in the study of cyclic groups, field extensions and cyclotomic polynomials of a finite field. v TABLE OF CONTENTS DECLARATION .......................................................................................................... ii DEDICATION ............................................................................................................. iii ACKNOWLEDEMENT .............................................................................................. iv ABSTRACT ...................................................................................................................v TABLE OF CONTENTS ............................................................................................. vi CHAPTER 1 ..................................................................................................................1 INRODUCTION ............................................................................................................1 1.1 BACKGROUND OF STUDY .............................................................................1 1.2 STATEMENT OF PROBLEM ............................................................................4 1.3 OBJECTIVE OF STUDY ....................................................................................5 1.4 METHODOLOGY ...............................................................................................5 1.5 JUSTIFICATION OF THE OBJECTIVES .........................................................5 1.6 ORGANIZATION OF THESIS ...........................................................................5 CHAPTER 2 ..................................................................................................................7 BASIC CNCEPTS .........................................................................................................7 2.1 Principle of Well-ordering ................................................................................7 2.1.1 Principle of Induction ...................................................................................8 2.1.3 Euclidean Algorithm....................................................................................10 2.1.4 The Division Algorithm..............................................................................11 2.2 FACTORS ..........................................................................................................11 2.2.1 Relatively Prime .........................................................................................13 2.2.2 Associates ....................................................................................................13 2.2.3 Prime element ..............................................................................................14 2.3 INTEGERS MODULO m ..................................................................................14 2.3.1 EULER’S PHI-FUNCTION ........................................................................15 2.3.2 Fermat's Little Theorem...............................................................................18 vi 2.3.3 Euler’s Theorem ..........................................................................................21 CHAPTER THREE .....................................................................................................23 GROUPS, RINGS AND FIELDS................................................................................23 3.1 GROUP ..................................................................................................................23 3.1.1 Axioms of a group ...........................................................................................23 3.1.3 Subgroup ......................................................................................................25 3.2 FINITE GROUP .................................................................................................28 3.2.1The Order of an Element ..............................................................................28 3.2.2 Lagrange Theorem .......................................................................................30 3.2.3 Cauchy Theorem ..........................................................................................31 3.3 CYCLIC GROUP ...............................................................................................32 3.3.1 Subgroup of Cyclic Groups .........................................................................33 3.4 HOMOMORPHISM ..........................................................................................34 3.4.1 Kernel of a Group ........................................................................................35 SOME APPLICATIONS OF EULER’S PHI-FUNCTION IN GROUP THEORY ...37 3.5 AUTOMORPHISMS OF A GROUP .................................................................37 3.5.1 Automorphism of Cyclic Group ..................................................................38 3.6 RING ......................................................................................................................41 3.6.1 Commutative Ring / A Ring with unity 1....................................................41 3.6.2 Subring.........................................................................................................42 3.6.3 Integral Domain ...........................................................................................42 3.7 FIELD.................................................................................................................44 3.7.1 Subfield ........................................................................................................44 3.7.2 Polynomials .................................................................................................46 CHAPTER 4 ................................................................................................................48 RING OF POLYNOMIALS ....................................................................................48 4.1 Ring of Polynomials .......................................................................................48 vii 4.1.1 Gauss theorem ...........................................................................................50 4.1.2 Gauss lemma................................................................................................51 4.1.3 Eisenstein's Irreducibility Criterion .............................................................51 4.2 Characteristic of a Field ..................................................................................53 4.3 Splitting Field .................................................................................................56 4.4 GALOIS GROUP ...............................................................................................59 4.5 CYCLOTOMIC POLYNOMIALS ....................................................................64 CHAPTER FIVE .........................................................................................................69 CONCLUSION AND RECCOMMENDATION ........................................................69 NOTATION .................................................................................................................70 REFERENCES ............................................................................................................71 viii CHAPTER 1 INRODUCTION A fundamental difficulty for beginning students is often the axiomatic nature of abstract algebra and the exacting need to follow the axioms precisely. To this end particular and extensive attention is paid to the integers, which are familiar objects of knowledge, and which, together with some simple properties of polynomials, are used to give