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Abstract Algebra: Monoids, Groups, Rings

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Abstract Algebra: Monoids, Groups, Rings Notes on Abstract Algebra John Perry University of Southern Mississippi [email protected] http://www.math.usm.edu/perry/ Copyright 2009 John Perry www.math.usm.edu/perry/ Creative Commons Attribution-Noncommercial-Share Alike 3.0 United States You are free: to Share—to copy, distribute and transmit the work • to Remix—to adapt the work Under• the following conditions: Attribution—You must attribute the work in the manner specified by the author or licen- • sor (but not in any way that suggests that they endorse you or your use of the work). Noncommercial—You may not use this work for commercial purposes. • Share Alike—If you alter, transform, or build upon this work, you may distribute the • resulting work only under the same or similar license to this one. With the understanding that: Waiver—Any of the above conditions can be waived if you get permission from the copy- • right holder. 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The best way to do this is with a link to this web page: http://creativecommons.org/licenses/by-nc-sa/3.0/us/legalcode Table of Contents Reference sheet for notation...........................................................iv A few acknowledgements..............................................................vi Preface ...............................................................................vii Overview ...........................................................................vii Three interesting problems ............................................................1 Part . Monoids 1. From integers to monoids ...........................................................4 1. Some facts about the integers .........................................................5 2. Integers, monomials, and monoids ...................................................11 3. Direct Products and Isomorphism ....................................................16 Part I. Groups 2. Groups ............................................................................23 1. Groups ...........................................................................23 2. The symmetries of a triangle.........................................................31 3. Cyclic groups and order ............................................................37 4. The roots of unity ..................................................................44 5. Elliptic Curves ....................................................................46 3. Subgroups .........................................................................50 1. Subgroups.........................................................................50 2. Cosets ............................................................................54 3. Lagrange’s Theorem ................................................................59 4. Quotient Groups ..................................................................62 5. “Clockwork” groups................................................................67 6. “Solvable” groups ..................................................................71 4. Isomorphisms ......................................................................76 1. Homomorphisms ..................................................................76 2. Consequences of isomorphism .......................................................82 3. The Isomorphism Theorem ..........................................................87 4. Automorphisms and groups of automorphisms.........................................91 5. Groups of permutations............................................................96 1. Permutations......................................................................96 2. Groups of permutations ...........................................................105 3. Dihedral groups ..................................................................107 4. Cayley’s Theorem .................................................................112 i 5. Alternating groups ................................................................115 6. The 15-puzzle ....................................................................120 6. Number theory ...................................................................123 1. GCD and the Euclidean Algorithm .................................................123 2. The Chinese Remainder Theorem ...................................................127 3. Multiplicative clockwork groups ....................................................134 4. Euler’s Theorem ..................................................................140 5. RSA Encryption ..................................................................144 Part II. Rings 7. Rings .............................................................................151 1. A structure for addition and multiplication ..........................................151 2. Integral Domains and Fields .......................................................154 3. Polynomial rings .................................................................159 4. Euclidean domains................................................................167 8. Ideals .............................................................................173 1. Ideals............................................................................173 2. Principal Ideals ...................................................................179 3. Prime and maximal ideals .........................................................182 4. Quotient Rings ...................................................................185 5. Finite Fields I ....................................................................190 6. Ring isomorphisms ...............................................................196 7. A generalized Chinese Remainder Theorem ..........................................201 8. Nullstellensatz....................................................................204 9. Rings and polynomial factorization................................................207 1. The link between factoring and ideals ...............................................207 2. Unique Factorization domains .....................................................210 3. Finite fields II ....................................................................213 4. Polynomial factorization in finite fields ..............................................218 5. Factoring integer polynomials ......................................................223 10. Gröbner bases ...................................................................228 1. Gaussian elimination ............................................................229 2. Monomial orderings .............................................................235 3. Matrix representations of monomial orderings ......................................242 4. The structure of a Gröbner basis ...................................................245 5. Buchberger’s algorithm ...........................................................255 6. Elementary applications ..........................................................263 11. Advanced methods of computing Gröbner bases ..................................268 1. The Gebauer-Möller algorithm ....................................................268 ii 2. The F4 algorithm ................................................................277 3. Signature-based algorithms to compute a Gröbner basis...............................282 Part III. Appendices Where can I go from here? ...........................................................291 Advanced group theory .............................................................291 Advanced ring theory ..............................................................291 Applications.......................................................................291 Hints to Exercises ...................................................................292 Hints to Chapter1 .................................................................292 Hints to Chapter2 .................................................................293 Hints to Chapter3 .................................................................294 Hints to Chapter4 .................................................................295 Hints to Chapter5 .................................................................296 Hints to Chapter6 .................................................................296 Hints to Chapter7 .................................................................297 Hints to Chapter8 .................................................................298 Hints to Chapter9 .................................................................299 Hints to Chapter 10 ................................................................299 Index................................................................................300 References...........................................................................304 iii Reference sheet for notation [r ] the element r + nZ of Zn g the group (or ideal) generated by g h i A3 the alternating group on three elements A/ G for G a group, A is a normal subgroup of G A/ R for R a ring, A is an ideal of R C
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