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. Other Rights—In no way are any of the following rights affected by the license: • Your fair dealing or fair use rights; ◦ Apart from the remix rights granted under this license, the author’s moral rights; ◦ Rights other persons may have either in the work itself or in how the work is used, ◦ such as publicity or privacy rights. Notice—For any reuse or distribution, you must make clear to others the license terms of • this work. 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 i Table of Contents Reference sheet for notation ...........................................................v A few acknowledgements .............................................................vii Preface ..............................................................................viii Overview ..........................................................................viii 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 ....................................................15 Part I. Groups 2. Groups ............................................................................22 1. Groups ...........................................................................22 2. The symmetries of a triangle.........................................................29 3. Cyclic groups and order ............................................................36 4. Elliptic Curves ....................................................................43 3. Subgroups .........................................................................47 1. Subgroups.........................................................................47 2. Cosets ............................................................................51 3. Lagrange’s Theorem ................................................................56 4. Quotient Groups ..................................................................59 5. “Clockwork” groups................................................................64 6. “Solvable” groups ..................................................................67 4. Isomorphisms ......................................................................72 1. From functions to isomorphisms .....................................................72 2. Consequences of isomorphism .......................................................78 3. The Isomorphism Theorem ..........................................................83 4. Automorphisms and groups of automorphisms.........................................87 5. Groups of permutations............................................................92 1. Permutations......................................................................92 2. Groups of permutations ...........................................................101 3. Dihedral groups ..................................................................103 4. Cayley’s Theorem .................................................................108 5. Alternating groups ................................................................111 ii 6. The 15-puzzle ....................................................................116 6. Number theory ...................................................................119 1. GCD and the Euclidean Algorithm .................................................119 2. The Chinese Remainder Theorem ...................................................123 3. Multiplicative clockwork groups ....................................................130 4. Euler’s Theorem ..................................................................136 5. RSA Encryption ..................................................................140 Part II. Rings 7. Rings .............................................................................147 1. A structure for addition and multiplication ..........................................147 2. Integral Domains and Fields .......................................................150 3. Polynomial rings .................................................................155 4. Euclidean domains................................................................163 8. Ideals .............................................................................169 1. Ideals............................................................................169 2. Principal Ideals ...................................................................175 3. Prime and maximal ideals .........................................................178 4. Quotient Rings ...................................................................181 5. Finite Fields I ....................................................................186 6. Ring isomorphisms ...............................................................192 7. A generalized Chinese Remainder Theorem ..........................................197 8. Nullstellensatz....................................................................200 9. Rings and polynomial factorization................................................203 1. The link between factoring and ideals ...............................................203 2. Unique Factorization domains .....................................................206 3. Finite fields II ....................................................................209 4. Polynomial factorization in finite fields ..............................................214 5. Factoring integer polynomials ......................................................219 10. Gröbner bases ...................................................................224 1. Gaussian elimination ............................................................225 2. Monomial orderings .............................................................231 3. Matrix representations of monomial orderings ......................................238 4. The structure of a Gröbner basis ...................................................241 5. Buchberger’s algorithm ...........................................................251 6. Elementary applications ..........................................................259 11. Advanced methods of computing Gröbner bases ..................................264 1. The Gebauer-Möller algorithm ....................................................264 2. The F4 algorithm ................................................................273 iii 3. Signature-based algorithms to compute a Gröbner basis...............................278 Part III. Appendices Where can I go from here? ...........................................................287 Advanced group theory .............................................................287 Advanced ring theory ..............................................................287 Applications.......................................................................287 Hints to Exercises ...................................................................288 Hints to Chapter1 .................................................................288 Hints to Chapter2 .................................................................288 Hints to Chapter3 .................................................................290 Hints to Chapter4 .................................................................291 Hints to Chapter5 .................................................................291 Hints to Chapter6 .................................................................292 Hints to Chapter7 .................................................................293 Hints to Chapter8 .................................................................294 Hints to Chapter9 .................................................................295 Hints to Chapter 10 ................................................................295 Index................................................................................296 References...........................................................................300 iv 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 the complex numbers a + b i : a, b C and i = p 1 [G,G] commutator subgroupf of a group