Introductory Abstract Algebras

1 Preliminaries 11
1.1 Well Ordering Principle . 19
1.3 Mathematical Induction . 24
1.4 Functions . 30

2 Basic Group Theory 39
2.1 De nition and Examples . 39
2.2 Basic Properties . 43

3 Subgroups 51
3.1 Examples . 51
3.2 Subgroup Tests . 53

4 Normal Subgroups and Lagrange Theorem 63
4.1 Cosets and their operations . 63
4.2 Normal Subgroup and Factor Group . 66

5 Homomorphisms and Isomorphisms 75
5.1 Homomorphisms . 75
5.2 Isomorphism Theorems . 80
5.3 Isomorphic In nite Groups . 86

6 Cyclic Groups 93
6.1 De nition and Examples . 93
6.2 Properties of Cyclic Groups . 94
6.3 Fundamental Theorem of Cyclic Groups . 98

7 Permutation Groups 107
7.1 De nition and Notations . 107
7.2 Properties of Permutations . 110
7.3 Alternating Groups . 113
7.4 Cayley's Theorem and Group Actions . 117

8 Automorphisms 127
8.1 Automorphisms of Cyclic Groups . 127
8.2 Inner Automorphisms . 129

9 Abelian Groups and Solvable Groups 137
9.1 Abelian Groups with Prime Orders . 137
9.2 Solvable Groups . 139

10 Products 147
10.1 External Direct Products . 147
10.2 Internal Direct Products . 149

11 Sylow Theorems 155
11.1 First Sylow Theorem . 155
11.2 Second Sylow Theorem and Simple Groups . 160
11.3 Third Sylow Theorem and Examples .162

12 Basic Ring Theory 169
12.1 De nition and Examples of Rings . 169
12.2 Polynomial Rings: New Ring From A Ring . 172
12.3 Properties of Ring Elements and Subrings . 175

13 More Rings 183
13.1 Integral Domains . 183
13.2 Fields . 185
13.3 Characteristics of Integral Domains . 189
13.4 Localizations of Integral Domains . 192

14 Polynomial Rings 199
14.1 Addition and Multiplication . 199
14.2 Division and Factors . 201
14.3 Zeros Of A Polynomial . 205
14.4 Formal Power Series Rings . 207

15 Ideals and Factor Rings 211
15.1 Ideals . 211
15.2 Quotient Rings . 214
15.3 Principal Ideal Domain (PID) . 217
15.4 Prime and Maximal Ideals . 220

16 Ring Homomorphisms 231
16.1 De nition . 232
16.2 Homomorphism and its Kernel . 236
16.3 Isomorphism Theorems . 242

17 Factorizations of Polynomials 249
17.1 Reducible and Irreducible Polynomials . 249
17.2 Tests for Irreducibility of Polynomials over Z . 252
17.3 Irreducible Polynomials and Fields . 256
17.4 Prime Numbers and Prime Polynomials . 260

18 Euclidean Domains 269
18.1 De nition and Examples . 269
18.2 Divisions In Rings . 272

19 Factorizations in an Integral Domain 281
19.1 De nitions and Examples . 281
19.2 ED, PID, & UFD . 284

20 Vector Spaces 289
20.1 De nition and Examples . 289
20.2 Subspaces and Spanning Sets . 295
20.3 Basis of a Vector Space . 300

21 Basic Field Theory: Prime Fields 311
21.1 Prime Fields . 311
21.2 Finite and In nite Fields . 313

22 Field Extensions Using a Zero of a Polynomial 319
22.1 Irreducible Polynomials and Extension Fields . 319
22.2 Zero of a Polynomial . 323
22.3 Splitting Field and Field Extensions . 326
22.4 Uniqueness of Splitting Fields . 330

23 Algebraic Extensions
23.1 Finite Extensions . 339
23.2 Minimal Polynomials . 343
23.3 Fundamental Theorem of Algebra . 350

24Geometric Constructions 355
24.1 Constructible Constructions . 355
24.2 Trisection . 358

25 Separable and Inseparable Extensions 363
25.1 Perfect Fields and Multiplicity of a Zero . 363
25.2 Separable Polynomials and Extensions . 365

26 Galois Theory 371
26.1 Normal Extensions . 371
26.2 Galois Extensions . 373
26.3 Fundamental Theorem of Galois Theory . 380

27 Appendix: Matrices and Their Determinants 387
27.1 Matrices . 387
27.2 Determinants . 392

Bibliography 397