Introductory Abstract Algebras

1 Preliminaries 9
1.1 Equivalence Relations 16
1.2 Mathematical Induction 22
1.3 Functions 27

2 Basic Group Theory 37
2.1 De nition and Examples 37
2.2 Basic Properties 41

3 Subgroups 49
3.1 Examples 49
3.2 Subgroup Tests 51

4 Normal Subgroups and Lagrange Theorem 61

5 Homomorphisms and Isomorphisms 71

6 Cyclic Groups 87

7 Permutation Groups 97

8 Automorphisms 115

9 Abelian Groups 125

10 Products 131

11 Sylow Theorems 137

12 Basic Ring Theory 153

13 More Rings 167

14 Polynomial Rings 179

15 Ideals and Factor Rings 189

16 Ring Homomorphisms 207

17 Factorizations of Polynomials 223

18 Euclidean Domains 241

19 Factorizations in an Integral Domain 251

20 Vector Spaces 259
20.1Vector Spaces 259
20.2 Subspaces 265
20.3 Basis of a Vector Space 270

21 Basic Field Theory 281

22 Field Extensions 291

23 Algebraic Extensions 307

24 Constructible Geometric Constructions 321

25 Separable and Inseparable Extensions 327

26 Galois Theory 333

27 Appendix: Matrices and Their Determinants 349
27.1 Matrices 349
27.2 Determinants 354

Bibliography 359