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BA Sethuraman California State University Northridge A GENTLE INTRODUCTION TO ABSTRACT ALGEBRA B.A. Sethuraman California State University Northridge ii Copyright © 2015 B.A. Sethuraman. Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version 1.3 or any later version published by the Free Software Foundation; with no Invariant Sec- tions, no Front-Cover Texts, and no Back-Cover Texts. A copy of the license is included in the section entitled \GNU Free Documentation License". Source files for this book are available at http://www.csun.edu/~asethura/GIAAFILES/GIAAV2.0-B/GIAAV2. 0-BSource.zip History 2012 Version 1.0. Created. Author B.A. Sethuraman. 2015 Version 2.0-B. Book Version. Author B.A. Sethuraman Contents To the Student: How to Read a Mathematics Book3 To the Student: Proofs9 1 Divisibility in the Integers 27 1.1 Further Exercises......................... 45 2 Rings and Fields 57 2.1 Rings: Definition and Examples................. 58 2.2 Subrings.............................. 83 2.3 Integral Domains and Fields................... 90 2.4 Ideals............................... 101 2.5 Quotient Rings.......................... 109 2.6 Ring Homomorphisms and Isomorphisms............ 117 2.7 Further Exercises......................... 137 3 Vector Spaces 165 3.1 Vector Spaces: Definition and Examples............ 166 3.2 Linear Independence, Bases, Dimension............. 177 3.3 Subspaces and Quotient Spaces................. 205 3.4 Vector Space Homomorphisms: Linear Transformations.... 216 3.5 Further Exercises......................... 236 iii iv CONTENTS 4 Groups 247 4.1 Groups: Definition and Examples................ 248 4.2 Subgroups, Cosets, Lagrange's Theorem............ 283 4.3 Normal Subgroups, Quotient Groups.............. 300 4.4 Group Homomorphisms and Isomorphisms........... 307 4.5 Further Exercises......................... 316 A Sets, Functions, and Relations 333 B Partially Ordered Sets, Zorn's Lemma 339 C GNU Free Documentation License 349 Index 361 List of Videos To the Student: Proofs: Exercise 0.10................. 23 To the Student: Proofs: Exercise 0.15................. 24 To the Student: Proofs: Exercise 0.19................. 25 To the Student: Proofs: Exercise 0.20................. 25 To the Student: Proofs: Exercise 0.21................. 25 Chapter1: GCD via Division Algorithm............... 46 Chapter1: Number of divisors of an integer............. 49 p Chapter1: 2 is not rational...................... 52 Chapter2: Binary operations on a set with n elements....... 59 Chapter2: Do the integers form a group with respect to multipli- cation?............................... 61 p Chapter2: When is a + b m zero?.................. 69 Chapter2: Does Z(2) contain 2=6?................... 69 Chapter2: Associativity of matrix addition............. 71 Chapter2: Direct product of matrix rings.............. 80 Chapter2: Proofs that some basic properties of rings hold..... 81 Chapter2: Additive identities of a ring and a subring........ 85 p p Chapter2: Are Q[ 2] and Z[ 2] fields?............... 94 Chapter2: Well-definedness of operations in Z=pZ ......... 98 Chapter2: Ideal of Z(2) ........................ 106 v vi LIST OF VIDEOS Chapter2: Ideal generated by a1, ::: , an ............... 107 Chapter2: Evaluation Homomorphism................ 126 p p Chapter2: Linear independence in Q[ 2; 3]............ 138 LIST OF VIDEOS 1 2 LIST OF VIDEOS To the Student: How to Read a Mathematics Book How should you read a mathematics book? The answer, which applies to every book on mathematics, and in particular to this one, can be given in one word|actively. You may have heard this before, but it can never be overstressed|you can only learn mathematics by doing mathematics. This means much more than attempting all the problems assigned to you (although attempting every problem assigned to you is a must). What it means is that you should take time out to think through every sentence and confirm every assertion made. You should accept nothing on trust; instead, not only should you check every statement, you should also attempt to go beyond what is stated, searching for patterns, looking for connections with other material that you may have studied, and probing for possible generalizations. Let us consider an example: Example 0.1 On page 65 in Chapter2, you will find the following sentence: Yet, even in this extremely familiar number system, mul- 3 4 How to Read a Mathematics Book tiplication is not commutative; for instance, ! ! ! ! 1 0 0 1 0 1 1 0 · 6= · : 0 0 0 0 0 0 0 0 (The \number system" referred to is the set of 2×2 matrices whose entries are real numbers.) When you read a sentence such as this, the first thing that you should do is verify the computation yourselves. Mathematical insight comes from mathematical experience, and you cannot expect to gain mathematical experience if you merely accept somebody else's word that the product on the left side of the equation does not equal the product on the right side. The very process of multiplying out these matrices will make the set of 2× 2 matrices a more familiar system of objects, but as you do the calculations, more things can happen if you keep your eyes and ears open. Some or all of the following may occur: 1. You may notice that not only are the two products not the same, but that the product on the right side gives you the zero matrix. This should make you realize that although it may seem impossible that two nonzero \numbers" can multiply out to zero, this is only because you are confining your thinking to the real or complex numbers. Already, the set of 2×2 matrices (with which you have at least some familiarity) contains nonzero elements whose product is zero. 2. Intrigued by this, you may want to discover other pairs of nonzero matrices that multiply out to zero. You will do this by taking arbitrary pairs of matrices and determining their product. It is quite probable that you will not find an appropriate pair. At this point you may be tempted to give up. However, you should not. You should try to be creative, and study how the entries in the various pairs of matrices you How to Read a Mathematics Book 5 have selected affect the product. It may be possible for you to change one or two entries in such a way that the product comes out to be zero. For instance, suppose you consider the product ! ! ! 1 1 4 0 6 0 · = 1 1 2 0 6 0 You should observe that no matter what the entries of the first matrix are, the product will always have zeros in the (1; 2) and the (2; 2) slots. This gives you some freedom to try to adjust the entries of the first matrix so that the (1; 1) and the (2; 1) slots also come out to be zero. After some experimentation, you should be able to do this. 3. You may notice a pattern in the two matrices that appear in our in- equality on page4. Both matrices have only one nonzero entry, and that entry is a 1. Of course, the 1 occurs in different slots in the two matrices. You may wonder what sorts of products occur if you take similar pairs of matrices, but with the nonzero 1 occuring at other lo- cations. To settle your curiosity, you will multiply out pairs of such matrices, such as ! ! 0 0 0 1 · ; 1 0 0 0 or ! ! 0 0 0 0 · : 1 0 1 0 You will try to discern a pattern behind how such matrices multiply. To help you describe this pattern, you will let ei;j stand for the matrix with 1 in the (i; j)-th slot and zeros everywhere else, and you will try to discover a formula for the product of ei;j and ek;l, where i, j, k, and l can each be any element of the set f1; 2g. 6 How to Read a Mathematics Book 4. You may wonder whether the fact that we considered only 2×2 matrices is significant when considering noncommutative multiplication or when considering the phenomenon of two nonzero elements that multiply out to zero. You will ask yourselves whether the same phenomena occur in the set of 3 × 3 matrices or 4 × 4 matrices. You will next ask yourselves whether they occur in the set of n × n matrices, where n is arbitrary. But you will caution yourselves about letting n be too arbitrary. Clearly n needs to be a positive integer, since \n × n matrices" is meaningless otherwise, but you will wonder whether n can be allowed to equal 1 if you want such phenomena to occur. 5. You may combine3 and4 above, and try to define the matrices ei;j analogously in the general context of n × n matrices. You will study the product of such matrices in this general context and try to discover a formula for their product. Notice that a single sentence can lead to an enormous amount of mathemati- cal activity! Every step requires you to be alert and actively involved in what you are doing. You observe patterns for yourselves, you ask yourselves ques- tions, and you try to answer these questions on your own. In the process, you discover most of the mathematics yourselves. This is really the only way to learn mathematics (and in particular, it is the way every professional math- ematician has learned the subject). Mathematical concepts are developed precisely because mathematicians observe patterns in various mathematical objects (such as the 2 × 2 matrices), and to have a good understanding of these concepts you must try to notice these patterns for yourselves. May you spend many many hours happily playing in the rich and beautiful world of mathematics! How to Read a Mathematics Book 7 Exercise 0.2 Carry out the program in steps (1) through (5) 8 How to Read a Mathematics Book To the Student: Proofs Many students confronting mathematics beyond elementary calculus for the first time are stumped at the idea of proofs.
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