INTRODUCTION TO ALGEBRA

By Y. HITAK

1 Introduction 1 What is mathematics? 1 Numbers 10 Elementary algebra 23 Sets 32 Modular Arithmetic 48 Matrices 52 Appendix: Logic 58 2 Rings 63 Rings and subrings 63 Homomorphisms and ideals 74 Factorisation 87 Fields 99 Appendix: Miscellany 103 3 Groups 110 Groups and subgroups 110 Subgroups and cosets 119 Homomorphisms and normal subgroups 124 Some special groups 132 Appendix: How many groups? 147 4 Vector spaces 150 Vector spaces and subspaces 150 Linear transformations and matrices 161 5 Modules 182 Introduction 182 Modules over a Euclidean domain 190 Applications 196 6 The number systems 209 To the complex numbers 210 Algebraic and transcendental numbers 220 More about sets 230 7 Further topics 237 Further group theory 237 Further ring theory 256 Further field theory 268 Other structures 278 8 Applications 299 Coding theory 299 Galois Theory 317 Further reading 334