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Fernando Q. Gouvêa Fernando Q AMS / MAA DOLCIANI MATHEMATICAL EXPOSITIONS VOL AMS / MAA DOLCIANI MATHEMATICAL EXPOSITIONS VOL 48 48 This book offers a concise overview of the theory of groups, rings, and Fields Rings, Groups, A Guide to and fields at the graduate level, emphasizing those aspects that are useful in other parts of mathematics. It focuses on the main ideas and how they hang together. It will be useful to both students and professionals. In addition to the standard material on groups, rings, modules, fields, and Galois theory, the book includes discussions of other important topics that are often omitted in the standard graduate course, including linear groups, group representations, the structure of Artinian rings, projective, injective and flat modules, Dedekind domains, and central simple algebras. All of the important theorems are discussed, without proofs but often with a discussion of the intui- tive ideas behind those proofs. Those looking for a way to review and refresh their basic algebra will benefit from reading this guide, and it will also serve as a ready reference for mathematicians who make use of algebra in their work. Fernando Q. Gouvêa Fernando Q. Gouvêa Q. Fernando AMSMAA / PRESS 4-Color Process 328 page • 5/8” • 50lb stock • finish size: 6” x 9” i i “master” — 2013/12/4 — 10:58 — page i — #1 i i 10.1090/dol/048 A Guide to Groups, Rings, and Fields i i i i i i “master” — 2013/12/4 — 10:58 — page ii — #2 i i c 2012 by The Mathematical Association of America (Incorporated) Library of Congress Catalog Card Number 2012950687 Print Edition ISBN 978-0-88385-355-9 Electronic Edition ISBN 978-1-61444-211-0 Printed in the United States of America Current Printing (last digit): 10987654321 i i i i i i “master” — 2013/12/4 — 10:58 — page iii — #3 i i The Dolciani Mathematical Expositions NUMBER FORTY-EIGHT MAA Guides # 8 A Guide to Groups, Rings, and Fields Fernando Q. Gouvˆea Colby College Published and Distributed by The Mathematical Association of America i i i i i i “master” — 2013/12/4 — 10:58 — page iv — #4 i i The DOLCIANI MATHEMATICAL EXPOSITIONS series of the Mathe- matical Association of America was established through a generous gift to the Association from Mary P. Dolciani, Professor of Mathematics at Hunter College of the City University of New York. In making the gift, Profes- sor Dolciani, herself an exceptionally talented and successful expositor of mathematics, had the purpose of furthering the ideal of excellence in math- ematical exposition. The Association, for its part, was delighted to accept the gracious ges- ture initiatingthe revolvingfund for this series from one who has served the Association with distinction, both as a member of the Committee on Pub- lications and as a member of the Board of Governors. It was with genuine pleasure that the Board chose to name the series in her honor. The books in the series are selected for their lucid expository style and stimulating mathematical content. Typically, they contain an ample supply of exercises, many with accompanying solutions. They are intended to be sufficiently elementary for the undergraduate and even the mathematically inclined high-school student to understand and enjoy, but also to be inter- esting and sometimes challenging to the more advanced mathematician. Committee on Books Frank Farris, Chair Dolciani Mathematical Expositions Editorial Board Underwood Dudley, Editor Jeremy S. Case Rosalie A. Dance Christopher Dale Goff Thomas M. Halverson Michael J. McAsey Michael J. Mossinghoff Jonathan Rogness Elizabeth D. Russell Robert W. Vallin i i i i i i “master” — 2013/12/4 — 10:58 — page v — #5 i i 1. Mathematical Gems, Ross Honsberger 2. Mathematical Gems II, Ross Honsberger 3. Mathematical Morsels, Ross Honsberger 4. Mathematical Plums, Ross Honsberger (ed.) 5. Great Moments in Mathematics (Before 1650), Howard Eves 6. Maxima and Minima without Calculus, Ivan Niven 7. Great Moments in Mathematics (After 1650), Howard Eves 8. Map Coloring, Polyhedra, and the Four-Color Problem, David Barnette 9. Mathematical Gems III, Ross Honsberger 10. More Mathematical Morsels, Ross Honsberger 11. Old and New Unsolved Problems in Plane Geometry and Number Theory, Victor Klee and Stan Wagon 12. Problems for Mathematicians, Young and Old, Paul R. Halmos 13. Excursions in Calculus: An Interplay of the Continuous and the Discrete, Robert M. Young 14. The Wohascum County Problem Book, George T. Gilbert, Mark Krusemeyer, and Loren C. Larson 15. Lion Hunting and Other Mathematical Pursuits: A Collection of Mathematics, Verse,and Stories by Ralph P.Boas, Jr., edited by Gerald L. Alexandersonand Dale H. Mugler 16. Linear Algebra Problem Book, Paul R. Halmos 17. From Erd˝os to Kiev: Problems of Olympiad Caliber, Ross Honsberger 18. Which Way Did the Bicycle Go? ...and Other Intriguing Mathematical Mys- teries, Joseph D. E. Konhauser, Dan Velleman, and Stan Wagon 19. In P´olya’s Footsteps: Miscellaneous Problems and Essays, Ross Honsberger 20. Diophantusand Diophantine Equations, I. G. Bashmakova(Updated by Joseph Silverman and translated by Abe Shenitzer) 21. Logic as Algebra, Paul Halmos and Steven Givant 22. Euler: The Master of Us All, William Dunham 23. The Beginningsand Evolution of Algebra, I. G. BashmakovaandG. S. Smirnova (Translated by Abe Shenitzer) 24. Mathematical Chestnuts from Around the World, Ross Honsberger 25. Counting on Frameworks:Mathematics to Aid the Design of Rigid Structures, Jack E. Graver 26. Mathematical Diamonds, Ross Honsberger 27. Proofs that Really Count: The Art of Combinatorial Proof, Arthur T. Benjamin and Jennifer J. Quinn 28. Mathematical Delights, Ross Honsberger 29. Conics, Keith Kendig 30. Hesiod’s Anvil: falling and spinning through heaven and earth, Andrew J. Simoson i i i i i i “master” — 2013/12/4 — 10:58 — page vi — #6 i i 31. A Garden of Integrals, Frank E. Burk 32. A Guide to Complex Variables (MAA Guides #1), Steven G. Krantz 33. Sink or Float? Thought Problems in Math and Physics, Keith Kendig 34. Biscuits of Number Theory, Arthur T. Benjamin and Ezra Brown 35. Uncommon Mathematical Excursions: Polynomia and Related Realms, Dan Kalman 36. When Less is More: Visualizing Basic Inequalities, Claudi Alsina and Roger B. Nelsen 37. A Guide to Advanced Real Analysis (MAA Guides #2), Gerald B. Folland 38. A Guide to Real Variables (MAA Guides #3), Steven G. Krantz 39. Voltaire’s Riddle: Microm´egas and the measure of all things, Andrew J. Simoson 40. A Guide to Topology,(MAA Guides #4), Steven G. Krantz 41. A Guide to Elementary Number Theory, (MAA Guides #5), Underwood Dud- ley 42. Charming Proofs: A Journey into Elegant Mathematics, Claudi Alsina and Roger B. Nelsen 43. Mathematics and Sports, edited by Joseph A. Gallian 44. A Guide to AdvancedLinear Algebra, (MAA Guides #6), Steven H. Weintraub 45. Icons of Mathematics: An Exploration of Twenty Key Images, Claudi Alsina and Roger B. Nelsen 46. A Guide to Plane Algebraic Curves, (MAA Guides #7), Keith Kendig 47. New Horizons in Geometry, Tom M. Apostol and Mamikon A. Mnatsakanian 48. A Guide to Groups,Rings, andFields, (MAA Guides #8), Fernando Q. Gouvˆea MAA Service Center P.O. Box 91112 Washington, DC 20090-1112 1-800-331-1MAA FAX: 1-301-206-9789 i i i i i i “master” — 2013/12/4 — 10:58 — page vii — #7 i i Contents Preface............................................ ............. xv A Guide to this Guide ........................................... 1 1 Algebra: Classical, Modern, and Ultramodern........... ..... 3 1.1 TheBeginningsofModernAlgebra . 4 1.2 ModernAlgebra....................... 6 1.3 UltramodernAlgebra . 7 1.4 WhatNext? ......................... 8 2 Categories........................................ .......... 9 2.1 Categories.......................... 9 2.2 Functors........................... 12 2.3 NaturalTransformations . 13 2.4 Products,Coproducts,andGeneralizations . 14 3 AlgebraicStructures............................... .......... 17 3.1 StructureswithOneOperation . 17 3.2 Rings ............................ 20 3.3 Actions ........................... 22 3.4 Semirings .......................... 24 3.5 Algebras........................... 25 3.6 OrderedStructures . 26 4 Groups andtheirRepresentations..................... ....... 29 4.1 Definitions.......................... 29 4.1.1 Groupsandhomomorphisms. 29 4.1.2 Subgroups ..................... 30 4.1.3 Actions....................... 31 4.1.4 G actingonitself..... ........ ..... 31 4.2 SomeImportantExamples . 32 4.2.1 Permutationgroups. 32 vii i i i i i i “master” — 2013/12/4 — 10:58 — page viii — #8 i i viii Contents 4.2.2 Symmetrygroups. 33 4.2.3 Otherexamples. 35 4.2.4 Topologicalgroups. 35 4.2.5 Freegroups..................... 36 4.3 ReframingtheDefinitions. 37 4.4 OrbitsandStabilizers . 38 4.4.1 Stabilizers ..................... 38 4.4.2 Orbits........................ 39 4.4.3 Actingbymultiplication . 40 4.4.4 Cosets ....................... 41 4.4.5 Countingcosetsandelements . 41 4.4.6 Doublecosets...... ....... ....... 42 4.4.7 Aniceexample..... ....... ....... 42 4.5 HomomorphismsandSubgroups . 44 4.5.1 Kernel,image,quotient . 44 4.5.2 Homomorphismtheorems . 46 4.5.3 Exactsequences . 48 4.5.4 H¨older’sdream. 48 4.6 ManyCheerfulSubgroups . 49 4.6.1 Generators,cyclicgroups . 49 4.6.2 Elementsoffiniteorder . 50 4.6.3 Finitelygenerated groups and the Burnsideproblem 51 4.6.4 Othernicesubgroups. 51 4.6.5 Conjugationandtheclassequation . 52 4.6.6 p-groups ...................... 53 4.6.7 Sylow’sTheoremandSylowsubgroups . 55 4.7 SequencesofSubgroups . 55 4.7.1 Compositionseries . 55 4.7.2 Central series, derived series,
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