Abstract Algebra: A Concrete Introduction

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Oktober 2000



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.


Introduction, Biography: Al-Kwharizmi. 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. 5. Fields. 6. Solvability by Radicals, Biography: Niels Henrik Abel. 7. Rings. 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. 18. Subgroups. 19. Subgroups Generated by Subsets. 20. Cosets. 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. APPENDICES. A. The Greek Alphabet. B. Proving Theorems, Biography: George Boole. C. Vector Spaces over Fields. D. Constructions with Straightedge and Compass.
EAN: 9780201437218
ISBN: 020143721X
Untertitel: Sprache: Englisch.
Erscheinungsdatum: Oktober 2000
Seitenanzahl: 512 Seiten
Format: kartoniert
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