# Book:Seth Warner/Modern Algebra

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## Seth Warner: Modern Algebra

Published $\text {1965}$, Dover Publications, Inc.

ISBN 0-486-66341-8

### Contents

Preface

CHAPTER $\text {I}$. ALGEBRAIC STRUCTURES
1. The Language of Set Theory
2. Compositions
3. Unions and Intersections of Sets
4. Neutral Elements and Inverses
5. Compositions and Inverses of Functions
6. Isomorphisms of Algebraic Structures
7. Semigroups and Groups

CHAPTER $\text {II}$. NEW STRUCTURES FROM OLD
8. Compositions Induced on Subsets
9. Compositions Induced on the Set of All Subsets
10. Equivalence Relations
11. Quotient Structures
12. Homomorphisms
13. Compositions Induced on Cartesian Products and Function Spaces

CHAPTER $\text {III}$. THE NATURAL NUMBERS
14. Orderings
15. Ordered Semigroups
16. The Natural Numbers
17. Finite Sets
18. Induced $N$-ary Operations
19. Combinatorial Analysis

CHAPTER $\text {IV}$. RINGS AND FIELDS
20. The Integers
21. Rings and Integral Domains
22. New Rings from Old
23. The Field of Rational Numbers
24. The Division Algorithm
25. Cyclic Groups and Lagrange's Theorem

CHAPTER $\text {V}$. VECTOR SPACES
26. Vector Spaces and Modules
27. Subspaces and Bases
28. Linear Transformations
29. Matrices
30. Linear Equations
31. Direct Sums and Quotient Spaces
32. Rings of Linear Operators

CHAPTER $\text {VI}$. POLYNOMIALS
33. Algebras
34. The Algebra of Polynomials
35. Principal Ideal Domains
36. Substitution
37. Irreducibility Criteria
38. Adjoining Roots
39. Finite Fields and Division Rings
40. Polynomials in Several Indeterminates

CHAPTER $\text {VII}$. THE REAL AND COMPLEX NUMBER FIELDS
41. Dedekind and Archimedean Ordered Fields
42. The Construction of a Dedekind Ordered Field
43. Isomorphisms of Archimedean Ordered Groups
44. The Field of Complex Numbers
45. The Algebra of Quaternions

CHAPTER $\text {VIII}$. ALGEBRAIC EXTENSIONS OF FIELDS
46. Algebraic Extensions
47. Constructions by Ruler and Compass
48. Galois Theory
49. Separable and Normal Extensions
50. Roots of Unity
51. Solving Quadratics, Cubics, and Quartics
52. Permutation Groups
53. Solving Polynomials by Radicals

CHAPTER $\text {IX}$. LINEAR OPERATORS
54. Diagonalizable Operators
55. Primary and Torsion-free Modules
56. Finitely Generated Modules
57. Decompositions of Linear Operators
58. Determinants

CHAPTER $\text {X}$. INNER PRODUCT SPACES
59. Inner Products
60. Orthonormal Bases
61. Adjoints
62. The Spectral Theorem
63. Linear Operators on Inner Product Spaces

CHAPTER $\text {XI}$. THE AXIOM OF CHOICE
64. The Axiom of Choice
65. Zorn's Lemma
66. Algebraic Closures

LIST OF SYMBOLS (PAGES 1-457)
LIST OF SYMBOLS (PAGES 459-797)

INDEX (PAGES 1-457)
INDEX (PAGES 459-797)

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## Critical View

Thorough, but many of the exercises are busy-work, and hence are not documented on $\mathsf{Pr} \infty \mathsf{fWiki}$.

## Source work progress

Second pass completed up to here
Missing result just before this:
Don't forget this:
Revisit this section:

Filling out exercises and examples (the worthwhile ones -- plenty are just makework):

These need completing:
• 1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text {II}$: New Structures from Old: $\S 13$: Compositions Induced on Cartesian Products and Function Spaces: Exercise $13.14 \ \text{(b)}$
• 1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text {II}$: New Structures from Old: $\S 13$: Compositions Induced on Cartesian Products and Function Spaces: Exercise $13.9 \ \text{(b)}$