A gentle introduction to abstract algebra
Sethuraman B.AContents
To the Student: How to Read a Mathematics Book 2
To the Student: Proofs 10
1 Divisibility in the Integers 35
1.1 Further Exercises . . . . . . . . . . . . . . . . . . . . . . . . 63
2 Rings and Fields 79
2.1 Rings: Definition and Examples . . . . . . . . . . . . . . . . 80
2.2 Subrings . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120
2.3 Integral Domains and Fields . . . . . . . . . . . . . . . . . . 131
2.4 Ideals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149
2.5 Quotient Rings . . . . . . . . . . . . . . . . . . . . . . . . . 161
2.6 Ring Homomorphisms and Isomorphisms . . . . . . . . . . . 173
2.7 Further Exercises . . . . . . . . . . . . . . . . . . . . . . . . 203
3 Vector Spaces 246
3.1 Vector Spaces: Definition and Examples . . . . . . . . . . . 247
3.2 Linear Independence, Bases, Dimension . . . . . . . . . . . . 264
3.3 Subspaces and Quotient Spaces . . . . . . . . . . . . . . . . 307
3.4 Vector Space Homomorphisms: Linear Transformations . . . 324
3.5 Further Exercises . . . . . . . . . . . . . . . . . . . . . . . . 355
4 Groups 370
4.1 Groups: Definition and Examples . . . . . . . . . . . . . . . 371
4.2 Subgroups, Cosets, Lagrange's Theorem . . . . . . . . . . . 423
4.3 Normal Subgroups, Quotient Groups . . . . . . . . . . . . . 450
4.4 Group Homomorphisms and Isomorphisms . . . . . . . . . . 460
4.5 Further Exercises . . . . . . . . . . . . . . . . . . . . . . . . 473
A Sets, Functions, and Relations 496
B Partially Ordered Sets, Zorn's Lemma 504
Index 517
C GNU Free Documentation License 523