An Introduction to Abstract Algebra ( De Gruyter Textbook )

Publication series ：De Gruyter Textbook

Author： Derek J.S. Robinson  

Publisher： De Gruyter‎

Publication year： 2003

E-ISBN: 9783110198164

P-ISBN(Paperback): 9783110175448

Subject： O15 algebra, number theory, combinatorial theory

Language： ENG

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Description

This is a high level introduction to abstract algebra which is aimed at readers whose interests lie in mathematics and in the information and physical sciences. In addition to introducing the main concepts of modern algebra, the book contains numerous applications, which are intended to illustrate the concepts and to convince the reader of the utility and relevance of algebra today. In particular applications to Polya coloring theory, latin squares, Steiner systems and error correcting codes are described. Another feature of the book is that group theory and ring theory are carried further than is often done at this level. There is ample material here for a two semester course in abstract algebra. The importance of proof is stressed and rigorous proofs of almost all results are given. But care has been taken to lead the reader through the proofs by gentle stages. There are nearly 400 problems, of varying degrees of difficulty, to test the reader's skill and progress. The book should be suitable for students in the third or fourth year of study at a North American university or in the second or third year at a university in Europe.

Chapter

1 Sets, relations and functions

pp.：  1 – 11

1.1 Sets and subsets

pp.：  11 – 11

1.2 Relations, equivalence relations and partial orders

pp.：  11 – 14

1.3 Functions

pp.：  14 – 19

1.4 Cardinality

pp.：  19 – 23

2 The integers

pp.：  23 – 27

2.1 Well-ordering and mathematical induction

pp.：  27 – 27

2.2 Division in the integers

pp.：  27 – 29

2.3 Congruences

pp.：  29 – 34

3 Introduction to groups

pp.：  34 – 41

3.1 Permutations of a set

pp.：  41 – 41

3.2 Binary operations: semigroups, monoids and groups

pp.：  41 – 49

3.3 Groups and subgroups

pp.：  49 – 54

4 Cosets, quotient groups and homomorphisms

pp.：  54 – 62

4.1 Cosets and Lagrange’s Theorem

pp.：  62 – 62

4.2 Normal subgroups and quotient groups

pp.：  62 – 70

4.3 Homomorphisms of groups

pp.：  70 – 77

5 Groups acting on sets

pp.：  77 – 88

5.2 Orbits and stabilizers

pp.：  88 – 91

5.1 Group actions and permutation representations

pp.：  88 – 88

5.3 Applications to the structure of groups

pp.：  91 – 95

5.4 Applications to combinatorics – counting labellings and graphs

pp.：  95 – 102

6 Introduction to rings

pp.：  102 – 109

6.1 Definition and elementary properties of rings

pp.：  109 – 109

6.2 Subrings and ideals

pp.：  109 – 113

6.3 Integral domains, division rings and fields

pp.：  113 – 117

7 Division in rings

pp.：  117 – 125

7.2 Principal ideal domains

pp.：  125 – 128

7.1 Euclidean domains

pp.：  125 – 125

7.3 Unique factorization in integral domains

pp.：  128 – 131

7.4 Roots of polynomials and splitting fields

pp.：  131 – 137

8 Vector spaces

pp.：  137 – 144

8.2 Linear independence, basis and dimension

pp.：  144 – 148

8.1 Vector spaces and subspaces

pp.：  144 – 144

8.3 Linear mappings

pp.：  148 – 157

8.4 Orthogonality in vector spaces

pp.：  157 – 165

9 The structure of groups

pp.：  165 – 173

9.1 The Jordan-Holder Theorem

pp.：  173 – 173

9.2 Solvable and nilpotent groups

pp.：  173 – 181

9.3 Theorems on finite solvable groups

pp.：  181 – 188

10 Introduction to the theory of fields

pp.：  188 – 195

10.1 Field extensions

pp.：  195 – 195

10.2 Constructions with ruler and compass

pp.：  195 – 200

10.3 Finite fields

pp.：  200 – 205

10.4 Applications to latin squares and Steiner triple systems

pp.：  205 – 209

11 Galois theory

pp.：  209 – 218

11.2 Automorphisms of field extensions

pp.：  218 – 223

11.1 Normal and separable extensions

pp.：  218 – 218

11.3 The Fundamental Theorem of Galois Theory

pp.：  223 – 231

11.4 Solvability of equations by radicals

pp.：  231 – 238

12 Further topics

pp.：  238 – 245

12.1 Zorn’s Lemma and its applications

pp.：  245 – 245

12.2 More on roots of polynomials

pp.：  245 – 250

12.3 Generators and relations for groups

pp.：  250 – 253

12.4 An introduction to error correcting codes

pp.：  253 – 264

Bibliography

pp.：  264 – 277

Index of notation

pp.：  277 – 279

Index

pp.：  279 – 283

LastPages

pp.：  283 – 293

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