Abstract Algebra and Solution by Radicals

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This advanced undergraduate-level text was recommended for teacher education by The American Mathematical Monthly and praised as a "most readable book." An ideal introduction to groups and Galois theory, it provides students with an appreciation of abstraction and arbitrary postulational systems, ideas that are central to automation.
The authors take the algebraic equation and the discovery of the insolubility of the quintic as their theme. Starting with treatments of groups, rings, fields, and polynomials, they advance to Galois theory, radicals and roots of unity, and solution by radicals. Thirteen appendixes supplement this volume, along with numerous examples, illustrations, commentaries, and exercises. Students who have completed a first-year college course in algebra or calculus will find it an accessible and well-written treatment. 

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Editorial Reviews

Reprint of the Saunders edition of 1971, now printed on acid-free paper. Annotation c. Book News, Inc., Portland, OR (booknews.com)
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Product Details

  • ISBN-13: 9780486477237
  • Publisher: Dover Publications
  • Publication date: 4/21/2010
  • Series: Dover Books on Mathematics Series
  • Pages: 224
  • Sales rank: 1,501,930
  • Product dimensions: 6.10 (w) x 9.20 (h) x 0.60 (d)

Table of Contents

How to Use This Book
The Negative Object of This Book
Ch. 1 Groups 1
Discussion of the Group Properties 3
Further Properties of Groups 5
Examples of Groups 9
Ch. 2 Other Abstract Algebras 18
Examples of Various Abstract Algebras 20
Countable Infinity 25
Equivalence and Other Relations 26
Isomorphism 33
Comparisons Among Abstract Algebras 43
Ch. 3 More About Groups 47
Permutations 49
Subgroups 59
Congruence and Cosets 62
Quotient Groups 70
Ch. 4 Mappings that Preserve Relations 78
Ch. 5 Groups of Prime Order; Two Alternating Groups 88
Historical Intermission 96
Some Comments on the Biographies 100
Ch. 6 Polynomials 104
Factoring Polynomials in I[x] 116
Ch. 7 Algebraic Field Extensions 125
Algebraic Numbers and Transcendental Numbers 126
Ch. 8 Galois Theory 142
Ch. 9 Radicals and Roots of Unity 150
Ch. 10 Solution by Radicals 155
Appendices 161
Appendix A - Q Has [actual symbol not reproducible] Members 163
Appendix B - R Has [actual symbol not reproducible] Members 164
Appendix C - C Has [actual symbol not reproducible] Members 166
Appendix D - Graphing Complex Numbers 167
Appendix E - A Lattice of Abstract Algebras 173
Appendix F - Modular Arithmetic 174
Appendix G - Graphical Group Representations 175
Appendix H - History of Mathematics 176
Appendix I - More About Field Extensions 177
Appendix J - More About Galois Theory 184
Appendix K - The Cyclic Group of nth Roots of Unity 189
Appendix L - Proof That a Solvable Equation Has a Solvable Galois Group 190
Appendix M - Impossible Constructions 194
Answers to Selected Exercises 199
Index 205
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