Abstract algebra: a concrete introduction
Langue : Anglais
Auteur : REDFIELD
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, Biography: Julia Robinson.
8. Ways in Which Polynomials Are Like the Integers.
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 Dyck.
16. Permutation Groups.
17. Group Homomorphisms, Biography: Arthur Cayley.
18. Subgroups, Biography: Leopold Kronecker.
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.
23. Normal Subgroups and Quotient Groups, Biography: Otto H lder.
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 Product of Groups, Biography: J. H. M. Wedderburn.
30. Finite Internal Direct Products of Groups.
31. Cauchy's Theorem and Groups of Order p
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, Biography: Julia Robinson.
8. Ways in Which Polynomials Are Like the Integers.
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 Dyck.
16. Permutation Groups.
17. Group Homomorphisms, Biography: Arthur Cayley.
18. Subgroups, Biography: Leopold Kronecker.
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.
23. Normal Subgroups and Quotient Groups, Biography: Otto H lder.
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 Product of Groups, Biography: J. H. M. Wedderburn.
30. Finite Internal Direct Products of Groups.
31. Cauchy's Theorem and Groups of Order p
2, Biography: Augustin-Louis Cauchy.
32. Abelian Groups with Prime Power Order.
33. The Fundamental Theorem of Finite Abelian Groups, Biography: Leopold Kronecker.
34. Dihedral Groups, Biography: Felix Klein.
35. The Sylow Theorems, Biography: Peter Ludvig Sylow.
36. Groups of Order Less than Sixteen.
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.
32. Abelian Groups with Prime Power Order.
33. The Fundamental Theorem of Finite Abelian Groups, Biography: Leopold Kronecker.
34. Dihedral Groups, Biography: Felix Klein.
35. The Sylow Theorems, Biography: Peter Ludvig Sylow.
36. Groups of Order Less than Sixteen.
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.
Date de parution : 02-2001
Ouvrage de 512 p.
Thème d’Abstract algebra: a concrete introduction :
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