Abstract Algebra: Theory and Applications

by Thomas W. Judson

Read here Read at LibreTextsPDF

This is an open source textbook designed to teach the principles and theory of abstract algebra to college juniors and seniors in a rigorous manner. Its strengths include a wide range of exercises, both computational and theoretical, plus many nontrivial applications. The first half of the book presents group theory, through the Sylow theorems. The second-half is suitable for a second semester and presents rings, integral domains, Boolean algebras, vector spaces, and fields, concluding with Galo

Description from LibreTexts.

Publisher
Stephen F. Austing State University
Subjects
Mathematics, Abstract and Geometric Algebra
Licence
GNU Free Documentation License, as LibreTexts states it
Who can read
Tweens profiles and older, judged from its title and subjects

Contents

LibreTexts writes equations in TeX, and here they show as TeX; the PDF has them typeset.

  1. Front Matter
  2. Acknowledgements
  3. TitlePage
  4. Colophon
  5. InfoPage
  6. Preface
  7. Table of Contents
  8. Licensing
  9. 1: Preliminaries
  10. 1.1: A Short Note on Proofs
  11. 1.2: Sets and Equivalence Relations
  12. 1.3: Reading Questions
  13. 1.4: Exercises
  14. 1.5: References and Suggested Readings
  15. 1.6: Sage
  16. 1.7: Sage Exercises
  17. 2: The Integers
  18. 2.1: Mathematical Induction
  19. 2.2: The Division Algorithm
  20. 2.3: Reading Questions
  21. 2.4: Exercises
  22. 2.5: Programming Exercises
  23. 2.6: References and Suggested Readings
  24. 2.7: Sage
  25. 2.8: Sage Exercises
  26. 3: Groups
  27. 3.4: Reading Questions
  28. 3.5: Exercises
  29. 3.6: Additional Exercises- Detecting Errors
  30. 3.7: References and Suggested Readings
  31. 3.8: Sage
  32. 3.9: Sage Exercises
  33. 3.1: Integer Equivalence Classes and Symmetries
  34. 3.2: Definitions and Examples
  35. 3.3: Subgroups
  36. 4: Cyclic Groups
  37. 4.1: Cyclic Subgroups
  38. 4.2: Multiplicative Group of Complex Numbers
  39. 4.3: The Method of Repeated Squares
  40. 4.4: Reading Questions
  41. 4.5: Exercises
  42. 4.6: Programming Exercises
  43. 4.7: References and Suggested Readings
  44. 4.8: Sage
  45. 4.9: Sage Exercises
  46. 5: Permutation Groups
  47. 5.1: Definitions and Notation
  48. 5.2: Dihedral Groups
  49. 5.3: Reading Questions
  50. 5.4: Exercises
  51. 5.5: Sage
  52. 5.6: Sage Exercises
  53. 6: Cosets and Lagrange's Theorem
  54. 6.1: Cosets
  55. 6.2: Lagrange's Theorem
  56. 6.3: Fermat's and Euler's Theorems
  57. 6.4: Reading Questions
  58. 6.5: Exercises
  59. 6.6: Sage
  60. 6.7: Sage Exercises
  61. 7: Introduction to Cryptography
  62. 7.1: Private Key Cryptography
  63. 7.2: Public Key Cryptography
  64. 7.3: Reading Questions
  65. 7.4: Exercises
  66. 7.5: Additional Exercises- Primality and Factoring
  67. 7.6: References and Suggested Readings
  68. 7.7: Sage
  69. 7.8: Sage Exercises
  70. 8: Algebraic Coding Theory
  71. 8.1: Sage Exercises
  72. 8.2: Error-Detecting and Correcting Codes
  73. 8.3: Linear Codes
  74. 8.4: Parity-Check and Generator Matrices
  75. 8.5: Efficient Decoding
  76. 8.6: Reading Questions
  77. 8.7: Exercises
  78. 8.8: Programming Exercises
  79. 8.9: References and Suggested Readings
  80. 8.10: Sage
  81. 9: Isomorphisms
  82. 9.1: Definition and Examples
  83. 9.2: Direct Products
  84. 9.3: Reading Questions
  85. 9.4: Exercises
  86. 9.5: Sage
  87. 9.6: Sage Exercises
  88. 10: Normal Subgroups and Factor Groups
  89. 10.1: Factor Groups and Normal Subgroups
  90. 10.2: The Simplicity of the Alternating Groups
  91. 10.3: Reading Questions
  92. 10.4: Exercises
  93. 10.5: Sage
  94. 10.6: Sage Exercises
  95. 11: Homomorphisms
  96. 11.1: Group Homomorphisms
  97. 11.2: The Isomorphism Theorms
  98. 11.3: Reading Questions
  99. 11.4: Exercises
  100. 11.5: Additional Exercises- Automorphisms
  101. 11.6: Sage
  102. 11.7: Sage Exercises
  103. 12: Matrix Groups and Symmetry
  104. 12.1: Matrix Groups
  105. 12.2: Symmetry
  106. 12.3: Reading Questions
  107. 12.4: Exercises
  108. 12.5: References and Suggested Readings
  109. 12.6: Sage
  110. 12.7: Sage Exercises
  111. 13: The Structure of Groups
  112. 13.1: Finite Abelian Groups
  113. 13.2: Solvable Groups
  114. 13.3: Reading Questions
  115. 13.4: Exercises
  116. 13.5: Programming Exercises
  117. 13.6: References and Suggested Readings
  118. 13.7: Sage
  119. 13.8: Sage Exercises
  120. 14: Group Actions
  121. 14.1: Groups Acting on Sets
  122. 14.2: The Class Equation
  123. 14.3: Burnside's Counting Theorem
  124. 14.4: Reading Questions
  125. 14.5: Exercises
  126. 14.6: Programming Exercise
  127. 14.7: References and Suggested Reading
  128. 14.8: Sage
  129. 14.9: Sage Exercises
  130. 15: The Sylow Theorems
  131. 15.1: The Sylow Theorems
  132. 15.2: Examples and Applications
  133. 15.3: Reading Questions
  134. 15.4: Exercises
  135. 15.5: A Project
  136. 15.6: References and Suggested Readings
  137. 15.7: Sage
  138. 15.8: Sage Exercises
  139. 16: Rings
  140. 16.2: Sage Exercises
  141. 16.3: Rings
  142. 16.4: Integral Domains and Fields
  143. 16.5: Ring Homomorphisms and Ideals
  144. 16.6: Maximal and Prime Ideals
  145. 16.7: An Application to Software Design
  146. 16.8: Reading Questions
  147. 16.9: Exercises
  148. 16.10: Programming Exercise
  149. 16.10: Sage
  150. 16.11: References and Suggested Readings
  151. 17: Polynomials
  152. 17.1: Polynomial Rings
  153. 17.2: The Division Algorithm
  154. 17.3: Irreducible Polynomials
  155. 17.4: Reading Questions
  156. 17.5: Exercises
  157. 17.6: Additional Exercises- Solving the Cubic and Quartic Equations
  158. 17.7: Sage
  159. 17.8: Sage Exercises
  160. 18: Integral Domains
  161. 18.1: Fields of Fractions
  162. 18.2: Factorization in Integral Domains
  163. 18.3: Reading Questions
  164. 18.4: Exercises
  165. 18.5: References and Suggested Readings
  166. 18.6: Sage
  167. 18.7: Sage Exercises
  168. 19: Lattices and Boolean Algebras
  169. 19.1: Lattices
  170. 19.2: Boolean Algebras
  171. 19.3: The Algebra of Electrical Circuits
  172. 19.4: Reading Questions
  173. 19.5: Exercises
  174. 19.6: Programming Exercises
  175. 19.7: References and Suggested Readings
  176. 19.8: Sage
  177. 19.9: Sage Exercises
  178. 20: Vector Spaces
  179. 20.1: Definitions and Examples
  180. 20.2: Subspaces
  181. 20.3: Linear Independence
  182. 20.4: Reading Questions
  183. 20.5: Exercises
  184. 20.6: References and Suggested Readings
  185. 20.7: Sage
  186. 20.8: Sage Exercises
  187. 21: Fields
  188. 21.1: Extension Fields
  189. 21.2: Splitting Fields
  190. 21.3: Geometric Constructions
  191. 21.4: Reading Questions
  192. 21.5: Exercises
  193. 21.6: References and Suggested Readings
  194. 21.7: Sage
  195. 21.8: Sage Exercise
  196. 22: Finite Fields
  197. 22.1: Structure of a Finite Field
  198. 22.2: Polynomial Codes
  199. 22.3: Reading Questions
  200. 22.4: Exercises
  201. 22.5: Additional Exercises- Error Correction for BCH Codes
  202. 22.6: References and Suggested Readings
  203. 22.7: Sage
  204. 22.8: Sage Exercises
  205. 23: Galois Theory
  206. 23.1: Field Automorphisms
  207. 23.2: The Fundamental Theorem
  208. 23.3: Applications
  209. 23.4: Reading Questions
  210. 23.5: Exercises
  211. 23.6: References and Suggested Readings
  212. 23.7: Sage
  213. 23.8: Sage Exercises
  214. Back Matter
  215. Index
  216. Detailed Licensing