ABSTRACT ALGEBRA WITH APPLICATIONS Irwin Kra, State University of New York at Stony Brook and University of California at Berkeley Contents Introduction 7 Standard Notation and Commonly Used Symbols 9 Chapter 1. The integers 11 1. Introduction 11 2. Induction 12 3. The division algorithm: gcd and lcm 19 4. Primes 29 5. The rationals, algebraic numbers and other beasts 34 5.1. The rationals, Q 34 5.2. The reals, R 35 5.3. The complex numbers, C 36 5.4. The algebraic numbers 36 5.5. The quaternions, H 36 6. Modular arithmetic 37 7. Solutions of linear congruences 44 8. Euler 50 9. Public key cryptography 55 10. A collection of beautiful results 57 Chapter 2. Foundations 59 1. Naive set theory 59 2. Functions 60 3. Relations 64 4. Order relations on Z and Q 67 4.1. Orders on Z 67 4.2. Orders on Q 68 5. The complex numbers 68 Chapter 3. Groups 71 1. Permutation groups 71 2. The order and sign of a permutation 77 3. Definitions and more examples of groups 83 Chapter 4. Group homomorphisms and isomorphisms. 95 1. Elementary group theory 95 2. Lagrange's theorem 98 3. Homomorphisms 100 4. Groups of small order 101 3 4 CONTENTS 4.1. G = 1 103 4.2. jGj = 2, 3, 5, 7 and, in fact, all primes 103 4.3. jGj = 4 103 4.4. jGj = 6 103 4.5. jGj = 8 104 5. Homomorphismsj j and quotients 106 6. Isomorphisms 110 6.1. Every group is a subgroup of a permutation group 110 6.2. Solvable groups 111 6.3. MORE sections to be included 111 Chapter 5. Algebraic structures 113 1. A collection of algebraic structures 113 2. The algebra of polynomials 118 2.1. The vector space of polynomials of degree n 120 2.2. The Euclidean algorithm (for polynomials) 120 2.3. Differentiation 124 3. Ideals 125 3.1. Ideals in commutative rings 125 3.2. Ideals in Z and C[x] 127 4. CRT revisited 128 5. Polynomials over more general fields 129 6. Fields of quotients and rings of rational functions 130 Chapter 6. Error correcting codes 131 1. ISBN 131 2. Groups and codes 131 Chapter 7. Roots of polynomials 143 1. Roots of polynomials 143 1.1. Derivatives and multiple roots 147 2. Circulant matrices 147 3. Roots of polynomials of small degree 152 3.1. Roots of linear and quadratic polynomials 153 3.2. The general case 154 3.3. Roots of cubics 155 3.4. Roots of quartics 156 3.5. Real roots and roots of absolute value 1 158 3.6. What goes wrong for polynomials of higher degree? 159 Chapter 8. Moduli for polynomials 161 1. Polynomials in three guises 161 2. An example from high school math: the quadratic polynomial 162 3. An equivalence relation 162 4. An example all high school math teachers should know: the cubic polynomial 164 5. Arbitrary real or complex polynomials 164 6. Back to the cubic polynomial 165 7. Standard forms for cubics 168 CONTENTS 5 8. Solving the cubic 170 9. Solving the quartic 171 10. Concluding remarks 172 11. A moduli (parameter) count 172 Chapter 9. Nonsolvability by radicals 175 1. Algebraic extensions of fields 175 2. Field embeddings 177 3. Splitting fields 178 4. Galois extensions 179 5. Quadratic, cubic and quartic extensions 179 5.1. Linear extensions 179 5.2. Quadratic extensions 179 5.3. Cubic extensions 179 5.4. Quartic extensions 180 6. Nonsolvability 180 Bibliography 183 Index 185 Introduction This book is closest in spirit to [7]. Except for Chapters 7 and 91, where the reader will need some results from linear algebra (which are reviewed), this book requires no formal mathematics prerequisites. Readers should, however, posses sufficient mathematical sophis- tication to appreciate a logical argument and what constitutes a proof. More than enough information on these topics can be found in [10]. The reader should be aware of the following features of the book that may not be stan- dard. I have cut the book down to a bare minimum. If a reader is interested in a given • chapter or it is part of a mathematics course, then every word in it should be read and understood. When requested all the details should be filled in and all exercises and problems done (their content may be needed in subsequent parts of the "main" text). At times I use a "familiar" concept before if is formally defined as in Example 1.3. • I use italics for terms defined, either formally in definitions or informally during a • proof or discussion. Most nontrivial calculations and nontrivial management of sets as well as certain • algebraic manipulations are performed using the symbolic manipulation programs MAPLE or MATHEMATICA. MAPLE and MATHEMATICA worksheets are included both in the text and on an • accompanying disc { this latter format will permit easy program modifications by the reader for further exploration and experimentation. This is not a text book on MAPLE nor on MATHEMATICA. See [3] for such a treatise. Rather, these pro- grams are used as tools to learn and do mathematics. I have tried to use only very simple MAPLE and MATHEMATICA programs and routines and to use, when- ever possible, commands that are similar to ordinary mathematical expressions and formulae. I have tried to keep a reasonable mixture between formal proofs and informality • (claims that certain statements are "obvious"). This book is an introduction to abstract algebra. I have particularly tried to pay attention to the needs of future high school mathematics teachers. With this in mind I have chosen applications such as public key cryptography and error correcting codes which use basic algebra as well as a study of polynomials and their roots which is such a big part of pre- college mathematics. Portions of the the material in this book were used as a basis for courses tought at Stony Brook and at Berkeley. The students challenged me with good questions and suggestions. I 1The tone and level of mathematical sophistication of these two chapters is considerably different in these two chapters from those in the others. Much more background is expected from the reader interested in these sections. 7 8 INTRODUCTION am very grateful to the students who read the material, corrected errors, and pointed out ways for improving the exposition. Errors, of course, remain and are the responsibility of the author. Standard Notation and Commonly Used Symbols A LIST OF SYMBOLS TERM MEANING Z integers Zn congruence classes of integers modulo n Zn∗ the units (invertible elements) in Zn Q rationals R reals C complex numbers a the absolute value of the number a j j gcd(a1; a2; :::; an) = (a1; a2; :::; an) the greatest common divisor of the integers a1, a2, ..., an lcm(a1; a2; :::; an) the least common multiple of the integers a1, a2, ..., an [a]n the congruence class modulo n containing the integer a { a square root of 1 z real part of the complex−number z <z imaginary part of the complex number z z ==x + {y x = z and y = z z¯ conjugate of<the complex=number z r = z absolute value of the complex number z θ = argj jz an argument of the complex number z z = reıθ r = z and θ = arg z R cardinalitj j y of set R j j Xcondition the set of x X that satisfy condition '(n) the Euler '-function2evaluated at the positive integer n ord[a]n the order of the congruence class [a]n a b the integer a divides the integer b j redn reduction of integers modulo n ker(θ) kernel of homomorphism θ Im(θ) image of homomorphism θ F ∗ the units (invertible elements) in the ring F R[x] polynomial ring over the commutative ring R F (α) smallest subfield of C containing F and α F (x) the field of rational functions for the field F 9 10 STANDARD NOTATION AND COMMONLY USED SYMBOLS STANDARD TERMINOLOGY TERM MEANING LHS left hand side elements of sets usually denoted by lower case letters sets usually denoted by upper case letters RHS right hand side iff if and only if proper subset ⊂ subset, may not be proper a ⊆A the element a is a member of the set A a 2 A the element a is not a member of the set A 62 the empty set A; the cardinality of the set A Aj jB the union of the sets A and B A [ B the intersection of the sets A and B A\c the complement of the set A A B A Bc − \ Xcondition the elements of X that satisfy condition CHAPTER 1 The integers All of us have been dealing with integers from a very young age. They have been studied by mathematicians for thousands of years. Yet much about them is unknown and, in their education, most people though they have consistently used integers have not paid much attention to their basic properties. Only in 2003 was it proven that it does not take too long to decide whether an integer is a prime or not. It is still unknown whether one can factor an integer (into its prime factors) in a reasonably short time; although the belief is that it cannot be done in what is called \polynomial time." It is also surprising, perhaps, that in addition to their obvious role in counting and recording of data, they have deep applications to everyday life. The next to the last section of the chapter descibes a public key encryption system that allows secure communication, (on the INTERNET, for example) that is based on a beautiful theorem of Euler and the fact that it is very hard to factor large integers; the last section contains a small collection of results that I found fascinating { some of them will be needed in subsequent chapters of the book.