This is a new text for the Abstract Algebra course. The author has written this text with a unique, yet historical, approach: solvability by radicals. This approach depends on a fields-first organization. However, professors wishing to commence their course with group theory will find that the Table of Contents is highly flexible, and contains a generous amount of group coverage.
Fields-first approach. Unique approach to subject matters, allows instructor to cover groups, rings and fields in one semester.
Flexible Table of Contents. Allows professors to teach the material in a traditional way, Groups-first.
Proof Technique Appendix. Offers tips and techniques to students to improve their proof writing skills.
Exercise sets. Large quantity of exercises for professors to choose from, with a varying degree of difficulty.
Historical perspective. Interesting, timely biographies of mathematicians.
Table of Contents I. PRELIMINARIES.
1. Properties of the Integers, Biography: Augustus de Morgan.
2. Solving Cubic and Quartic Polynomial Equations, Historical Note: How the Cubic and Quartic Equations were Solved.
3. Complex Numbers. Historical Note: Highlights in the Development of the Complex Numbers.
4. Some Other Examples, Biography: William Rowan Hamilton.
II. ALGEBRAIC EXTENSION FIELDS.
6. Solvability by Radicals, Biography: Niels Henrik Abel.
8. Ways in Which Polynomials Are Like the Integers, Biography: Julia Robinson.
9. Principal Ideals, Biography: Emmy Noether.
10. Algebraic Elements.
11. Eisenstein's Irreducibility Criterion, Biography: Gotthold Eisenstein.
12. Extension Fields as Vector Spaces.
13. Automorphisms of Fields, Biography: Evariste Galois.
14. Counting Automorphisms, Biography: Richard Dedekind.
III. ELEMENTARY GROUP THEORY.
15. Groups, Biography: Walther von Dyck.
16. Permutation Groups.
17. Group Homomorphisms, Biography: Arthur Cayley.
19. Subgroups Generated by Subsets.
21. Finite Groups and Lagrange's Theorem, Biography: Joseph Louis Lagrange.
22. Equivalence Relations and Cauchy's Theorem, Biography: Augustin-Louis Cauchy.
23. Normal Subgroups and Quotient Groups, Biography: Otto Holder.
24. The Homomorphism Theorem for Groups, Biography: B. L. van derWaerden.
IV. POLYNOMIAL EQUATIONS NOT SOLVABLE BY RADICALS.
25. Galois Groups of Radical Extensions.
26. Solvable Groups and Commutator Subgroups, Biography: William Burnside.
27. Solvable Galois Groups.
28. Polynomial Equations Not Solvable by Radicals, Biography: Paolo Ruffini.
V. FINITE GROUPS.
29. Finite External Direct Products of Groups, Biography: J. H. M. Wedderburn.
30. Finite Internal Direct Products of Groups.
31. Abelian Groups with Prime Power Order.
32. The Fundamental Theorem of Finite Abelian Groups, Biography: Leopold Kronecker.
33. Dihedral Groups, Biography: Felix Klein.
34. Cauchy's Theorem.
35. The Sylow Theorems, Biography: Peter Ludvig Sylow.
36. Groups of Order Less than 16.
37. Groups of Even Permuatations, Biography: Camille Jordan.
38. Semidirect Products.