Abstract Algebra Theory and Applications Thomas W Judson Stephen F Austin State University Sage Exercises for Abstract Algebra Robert A Beezer University of Puget Sound July 10, 2019 Edition: Annual Edition 2019 Website: abstract.pugetsound.edu ©1997–2019 Thomas W Judson, Robert A Beezer Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version 1.2 or any later version published by the Free Software Foundation; with no Invariant Sections, no Front-Cover Texts, and no Back-Cover Texts A copy of the license is included in the appendix entitled “GNU Free Documentation License.” Acknowledgements I would like to acknowledge the following reviewers for their helpful comments and suggestions • David Anderson, University of Tennessee, Knoxville • Robert Beezer, University of Puget Sound • Myron Hood, California Polytechnic State University • Herbert Kasube, Bradley University • John Kurtzke, University of Portland • Inessa Levi, University of Louisville • Geoffrey Mason, University of California, Santa Cruz • Bruce Mericle, Mankato State University • Kimmo Rosenthal, Union College • Mark Teply, University of Wisconsin I would also like to thank Steve Quigley, Marnie Pommett, Cathie Griffin, Kelle Karshick, and the rest of the staff at PWS Publishing for their guidance throughout this project It has been a pleasure to work with them Robert Beezer encouraged me to make Abstract Algebra: Theory and Applications available as an open source textbook, a decision that I have never regretted With his assistance, the book has been rewritten in PreTeXt (pretextbook.org), making it possible to quickly output print, web, pdf versions and more from the same source The open source version of this book has received support from the National Science Foundation (Awards #DUE-1020957, #DUE–1625223, and #DUE–1821329) v vi Preface This text is intended for a one or two-semester undergraduate course in abstract algebra Traditionally, these courses have covered the theoretical aspects of groups, rings, and fields However, with the development of computing in the last several decades, applications that involve abstract algebra and discrete mathematics have become increasingly important, and many science, engineering, and computer science students are now electing to minor in mathematics Though theory still occupies a central role in the subject of abstract algebra and no student should go through such a course without a good notion of what a proof is, the importance of applications such as coding theory and cryptography has grown significantly Until recently most abstract algebra texts included few if any applications However, one of the major problems in teaching an abstract algebra course is that for many students it is their first encounter with an environment that requires them to rigorous proofs Such students often find it hard to see the use of learning to prove theorems and propositions; applied examples help the instructor provide motivation This text contains more material than can possibly be covered in a single semester Certainly there is adequate material for a two-semester course, and perhaps more; however, for a one-semester course it would be quite easy to omit selected chapters and still have a useful text The order of presentation of topics is standard: groups, then rings, and finally fields Emphasis can be placed either on theory or on applications A typical one-semester course might cover groups and rings while briefly touching on field theory, using Chapters through 6, 9, 10, 11, 13 (the first part), 16, 17, 18 (the first part), 20, and 21 Parts of these chapters could be deleted and applications substituted according to the interests of the students and the instructor A two-semester course emphasizing theory might cover Chapters through 6, 9, 10, 11, 13 through 18, 20, 21, 22 (the first part), and 23 On the other hand, if applications are to be emphasized, the course might cover Chapters through 14, and 16 through 22 In an applied course, some of the more theoretical results could be assumed or omitted A chapter dependency chart appears below (A broken line indicates a partial dependency.) vii viii Chapters 1–6 Chapter Chapter Chapter Chapter 10 Chapter 11 Chapter 13 Chapter 16 Chapter 12 Chapter 17 Chapter 18 Chapter 20 Chapter 14 Chapter 15 Chapter 19 Chapter 21 Chapter 22 Chapter 23 Though there are no specific prerequisites for a course in abstract algebra, students who have had other higher-level courses in mathematics will generally be more prepared than those who have not, because they will possess a bit more mathematical sophistication Occasionally, we shall assume some basic linear algebra; that is, we shall take for granted an elementary knowledge of matrices and determinants This should present no great problem, since most students taking a course in abstract algebra have been introduced to matrices and determinants elsewhere in their career, if they have not already taken a sophomore or junior-level course in linear algebra Exercise sections are the heart of any mathematics text An exercise set appears at the end of each chapter The nature of the exercises ranges over several categories; computational, conceptual, and theoretical problems are included A section presenting hints and solutions to many of the exercises appears at the end of the text Often in the solutions a proof is only sketched, and it is up to the student to provide the details The exercises range in difficulty from very easy to very challenging Many of the more substantial problems require careful thought, so the student should not be discouraged if the solution is not forthcoming after a few minutes of work There are additional exercises or computer projects at the ends of many of the chapters The computer projects usually require a knowledge of programming All of these exercises and projects are more substantial in nature and allow the exploration of new results and theory Sage (sagemath.org) is a free, open source, software system for advanced mathematics, which is ideal for assisting with a study of abstract algebra Sage can be used either on your own computer, a local server, or on CoCalc (cocalc.com) Robert Beezer has written a comprehensive ix introduction to Sage and a selection of relevant exercises that appear at the end of each chapter, including live Sage cells in the web version of the book All of the Sage code has been subject to automated tests of accuracy, using the most recent version available at this time: SageMath Version 8.8 (released 2019-07-02) Thomas W Judson Nacogdoches, Texas 2019 x Contents Acknowledgements v Preface vii Preliminaries 1.1 A Short Note on Proofs 1.2 Sets and Equivalence Relations 1.3 Exercises 1.4 References and Suggested Readings 1 14 16 The 2.1 2.2 2.3 2.4 2.5 17 17 20 24 26 27 Groups 3.1 Integer Equivalence Classes and Symmetries 3.2 Definitions and Examples 3.3 Subgroups 3.4 Exercises 3.5 Additional Exercises: Detecting Errors 3.6 References and Suggested Readings 29 29 33 38 40 43 45 Cyclic Groups 4.1 Cyclic Subgroups 4.2 Multiplicative Group of Complex Numbers 4.3 The Method of Repeated Squares 4.4 Exercises 4.5 Programming Exercises 4.6 References and Suggested Readings 47 47 50 54 56 59 59 Permutation Groups 5.1 Definitions and Notation 5.2 Dihedral Groups 5.3 Exercises 61 61 68 72 Integers Mathematical Induction The Division Algorithm Exercises Programming Exercises References and Suggested Readings xi xii Cosets and Lagrange’s Theorem 6.1 Cosets 6.2 Lagrange’s Theorem 6.3 Fermat’s and Euler’s Theorems 6.4 Exercises 77 77 79 81 82 Introduction to Cryptography 7.1 Private Key Cryptography 7.2 Public Key Cryptography 7.3 Exercises 7.4 Additional Exercises: Primality and Factoring 7.5 References and Suggested Readings 85 86 88 91 92 94 Algebraic Coding Theory 8.1 Error-Detecting and Correcting Codes 8.2 Linear Codes 8.3 Parity-Check and Generator Matrices 8.4 Efficient Decoding 8.5 Exercises 8.6 Programming Exercises 8.7 References and Suggested Readings 95 95 102 105 110 113 117 117 Isomorphisms 119 9.1 Definition and Examples 119 9.2 Direct Products 123 9.3 Exercises 126 10 Normal Subgroups and Factor Groups 131 10.1 Factor Groups and Normal Subgroups 131 10.2 The Simplicity of the Alternating Group 133 10.3 Exercises 136 11 Homomorphisms 11.1 Group Homomorphisms 11.2 The Isomorphism Theorems 11.3 Exercises 11.4 Additional Exercises: Automorphisms 139 139 141 144 145 12 Matrix Groups and Symmetry 12.1 Matrix Groups 12.2 Symmetry 12.3 Exercises 12.4 References and Suggested Readings 147 147 154 160 162 13 The 13.1 13.2 13.3 13.4 13.5 165 165 169 173 174 174 Structure of Groups Finite Abelian Groups Solvable Groups Exercises Programming Exercises References and Suggested Readings 345 equation are a = ±1, b = 18.3.2 Hint (a) = −i(1 + 2i)(2 + i); (c) + 8i = −i(1 + i)2 (2 + i)2 18.3.4 Hint True 18.3.9 Hint z/w ∈ Q(i) Let z = a + bi and w = c + di ̸= be in Z[i] Prove that 18.3.15 Hint Let a = ub with u a unit Then ν(b) ≤ ν(ub) ≤ ν(a) Similarly, ν(a) ≤ ν(b) 18.3.16 Hint Show that 21 can be factored in two different ways 19 · Lattices and Boolean Algebras 19.4 · Exercises 19.4.2 Hint 30 10 15 19.4.5 Hint False 19.4.6 Hint (a) (a ∨ b ∨ a′ ) ∧ a a a b a′ (c) a ∨ (a ∧ b) a b a 19.4.8 Hint Not equivalent 19.4.10 Hint (a) a′ ∧ [(a ∧ b′ ) ∨ b] = a ∧ (a ∨ b) 19.4.14 Hint Let I, J be ideals in R We need to show that I + J = {r + s : r ∈ I and s ∈ J} is the smallest ideal in R containing both I and J If r1 , r2 ∈ I and s1 , s2 ∈ J, then (r1 +s1 )+(r2 +s2 ) = (r1 +r2 )+(s1 +s2 ) is in I + J For a ∈ R, a(r1 + s1 ) = ar1 + as1 ∈ I + J; hence, I + J is an ideal in R 19.4.18 Hint (a) No 346APPENDIX B HINTS AND ANSWERS TO SELECTED EXERCISES 19.4.20 Hint (⇒) a = b ⇒ (a ∧ b′ ) ∨ (a′ ∧ b) = (a ∧ a′ ) ∨ (a′ ∧ a) = O ∨ O = O (⇐) (a ∧ b′ ) ∨ (a′ ∧ b) = O ⇒ a ∨ b = (a ∨ a) ∨ b = a ∨ (a ∨ b) = a ∨ [I ∧ (a ∨ b)] = a ∨ [(a ∨ a′ ) ∧ (a ∨ b)] = [a ∨ (a ∧ b′ )] ∨ [a ∨ (a′ ∧ b)] = a ∨ [(a ∧ b′ ) ∨ (a′ ∧ b)] = a ∨ = a A symmetric argument shows that a ∨ b = b 20 · Vector Spaces 20.4 · Exercises √ √ √ √ √ 20.4.3 Hint Q( 2, ) has basis {1, 2, 3, } over Q 20.4.5 Hint The set {1, x, x2 , , xn−1 } is a basis for Pn 20.4.7 Hint (a) Subspace of dimension with basis {(1, 0, −3), (0, 1, 2)}; (d) not a subspace 20.4.10 Hint Since = α0 = α(−v + v) = α(−v) + αv, it follows that −αv = α(−v) 20.4.12 Hint Let v0 = 0, v1 , , ∈ V and α0 ̸= 0, α1 , , αn ∈ F Then α0 v0 + · · · + αn = 20.4.15 Linear Transformations Hint (a) Let u, v ∈ ker(T ) and α ∈ F Then T (u + v) = T (u) + T (v) = T (αv) = αT (v) = α0 = Hence, u + v, αv ∈ ker(T ), and ker(T ) is a subspace of V (c) The statement that T (u) = T (v) is equivalent to T (u − v) = T (u) − T (v) = 0, which is true if and only if u − v = or u = v 20.4.17 Direct Sums Hint (a) Let u, u′ ∈ U and v, v ′ ∈ V Then (u + v) + (u′ + v ′ ) = (u + u′ ) + (v + v ′ ) ∈ U + V α(u + v) = αu + αv ∈ U + V 21 · Fields 21.4 · Exercises (a) x4 − (2/3)x2 − 62/9; (c) x4 − 2x2 + 25 √ √ √ √ √ 21.4.2 Hint (a) {1, 2, 3, }; (c) {1, i, 2, i}; (e) {1, 21/6 , 21/3 , 21/2 , 22/3 , 25/6 } √ √ 21.4.3 Hint (a) Q( 3, ) 21.4.1 Hint 21.4.5 Hint Use the fact that the elements of Z2 [x]/⟨x3 +x+1⟩ are 0, 1, α, + α, α2 , + α2 , α + α2 , + α + α2 and the fact that α3 + α + = 21.4.8 Hint False 21.4.14 Hint Suppose that E is algebraic over F and K is algebraic over E Let α ∈ K It suffices to show that α is algebraic over some finite extension of F Since α is algebraic over E, it must be the zero of some polynomial p(x) = β0 + β1 x + · · · + βn xn in E[x] Hence α is algebraic over F (β0 , , βn ) √ √ √ √ √ 21.4.22 Hint Since } is √a basis for Q( 3,√ )√over √ √ √ {1, √ 3, 7, 21√ Q, Q( 3, ) ⊃ Q( 3+ ) Since [Q( 3, ) : Q] = 4, [Q(√ 3+ √7 ) : Q] = √ or√4 Since √ the degree of the minimal polynomial of + is √ 4, Q( 3, ) = Q( + ) 347 21.4.27 Hint Let β ∈ F (α) not in F Then β = p(α)/q(α), where p and q are polynomials in α with q(α) ̸= and coefficients in F If β is algebraic over F , then there exists a polynomial f (x) ∈ F [x] such that f (β) = Let f (x) = a0 + a1 x + · · · + an xn Then ( ) ( ) ( )n p(α) p(α) p(α) = f (β) = f = a0 + a1 + · · · + an q(α) q(α) q(α) Now multiply both sides by q(α)n to show that there is a polynomial in F [x] that has α as a zero 21.4.28 Hint See the comments following Theorem 21.13, p 277 22 · Finite Fields 22.3 · Exercises 22.3.1 Hint Make sure that you have a field extension 22.3.4 Hint There are eight elements in Z2 (α) Exhibit two more zeros of x3 + x2 + other than α in these eight elements 22.3.5 Hint Find an irreducible polynomial p(x) in Z3 [x] of degree and show that Z3 [x]/⟨p(x)⟩ has 27 elements 22.3.7 Hint (a) x5 − = (x + 1)(x4 + x3 + x2 + x + 1); (c) x9 − = (x + 1)(x2 + x + 1)(x6 + x3 + 1) 22.3.8 Hint True 22.3.11 Hint (a) Use the fact that x7 − = (x + 1)(x3 + x + 1)(x3 + x2 + 1) 22.3.12 Hint False 22.3.17 Hint If p(x) ∈ F [x], then p(x) ∈ E[x] 22.3.18 Hint Since α is algebraic over F of degree n, we can write any element β ∈ F (α) uniquely as β = a0 + a1 α + · · · + an−1 αn−1 with ∈ F There are q n possible n-tuples (a0 , a1 , , an−1 ) 22.3.24 Wilson’s Theorem Hint Factor xp−1 − over Zp 23 · Galois Theory 23.4 · Exercises 23.4.1 Hint (a) Z2 ; (c) Z2 × Z2 × Z2 23.4.2 Hint (a) Separable over Q since x3 + 2x2 − x − = (x − 1)(x + 1)(x + 2); (c) not separable over Z3 since x4 + x2 + = (x + 1)2 (x + 2)2 23.4.3 Hint If [GF(729) : GF(9)] = [GF(729) : GF(3)]/[GF(9) : GF(3)] = 6/2 = 3, then G(GF(729)/ GF(9)) ∼ = Z3 A generator for G(GF(729)/ GF(9)) is σ, where σ36 (α) = α3 = α729 for α ∈ GF(729) 23.4.4 Hint (a) S5 ; (c) S3 ; (g) see Example 23.10, p 312 23.4.5 Hint (a) Q(i) 23.4.7 Hint Let E be the splitting field of a cubic polynomial in F [x] Show that [E : F ] is less than or equal to and is divisible by Since 348APPENDIX B HINTS AND ANSWERS TO SELECTED EXERCISES G(E/F ) is a subgroup of S3 whose order is divisible by 3, conclude that this group must be isomorphic to Z3 or S3 23.4.9 Hint G is a subgroup of Sn 23.4.16 Hint True 23.4.20 Hint (a) Clearly ω, ω , , ω p−1 are distinct since ω ̸= or To show that ω i is a zero of Φp , calculate Φp (ω i ) (b) The conjugates of ω are ω, ω , , ω p−1 Define a map ϕi : Q(ω) → Q(ω i ) by ϕi (a0 + a1 ω + · · · + ap−2 ω p−2 ) = a0 + a1 ω i + · · · + cp−2 (ω i )p−2 , where ∈ Q Prove that ϕi is an isomorphism of fields Show that ϕ2 generates G(Q(ω)/Q) (c) Show that {ω, ω , , ω p−1 } is a basis for Q(ω) over Q, and consider which linear combinations of ω, ω , , ω p−1 are left fixed by all elements of G(Q(ω)/Q) C Notation The following table defines the notation used in this book Page numbers or references refer to the first appearance of each symbol Symbol a∈A N Z Q R C A⊂B ∅ A∪B A∩B A′ A\B A×B An id f −1 a ≡ b (mod n) n! ( ) n k a|b gcd(a, b) P(X) lcm(m, n) Zn U (n) Mn (R) det A GLn (R) Q8 C∗ Description Page a is in the set A the natural numbers the integers the rational numbers the real numbers the complex numbers A is a subset of B the empty set the union of sets A and B the intersection of sets A and B complement of the set A difference between sets A and B Cartesian product of sets A and B A × · · · × A (n times) identity mapping 10 inverse of the function f 10 a is congruent to b modulo n 13 n factorial 18 binomial coefficient n!/(k!(n − k)!) 18 a divides b 20 greatest common divisor of a and b 20 power set of X 24 the least common multiple of m and n 25 the integers modulo n 29 group of units in Zn 35 the n × n matrices with entries in R 35 the determinant of A 35 the general linear group 35 the group of quaternions 36 the multiplicative group of complex num36 bers (Continued on next page) 349 350 APPENDIX C NOTATION Symbol Description |G| R∗ Q∗ SLn (R) Z(G) ⟨a⟩ |a| cis θ T Sn (a1 , a2 , , ak ) An Dn [G : H] LH the order of a group 36 the multiplicative group of real numbers 38 the multiplicative group of rational numbers 38 the special linear group 38 the center of a group 43 cyclic group generated by a 47 the order of an element a 48 cos θ + i sin θ 52 the circle group 53 the symmetric group on n letters 61 cycle of length k 63 the alternating group on n letters 67 the dihedral group 68 index of a subgroup H in a group G 78 the set of left cosets of a subgroup H in a 79 group G the set of right cosets of a subgroup H in a 79 group G a does not divide b 81 Hamming distance between x and y 100 the minimum distance of a code 100 the weight of x 100 the set of m × n matrices with entries in Z2 104 null space of a matrix H 104 Kronecker delta 108 G is isomorphic to a group H 119 automorphism group of a group G 128 ig (x) = gxg −1 129 inner automorphism group of a group G 129 right regular representation 129 factor group of G mod N 132 commutator subgroup of G 137 kernel of ϕ 140 matrix 148 orthogonal group 150 length of a vector x 150 special orthogonal group 153 Euclidean group 153 orbit of x 176 fixed point set of g 177 isotropy subgroup of x 177 normalizer of s subgroup H 190 the ring of quaternions 201 the Gaussian integers 203 characteristic of a ring R 204 ring of integers localized at p 216 degree of a polynomial 220 ring of polynomials over a ring R 220 (Continued on next page) RH a∤b d(x, y) dmin w(x) Mm×n (Z2 ) Null(H) δij G∼ =H Aut(G) ig Inn(G) ρg G/N G′ ker ϕ (aij ) O(n) ∥x∥ SO(n) E(n) Ox Xg Gx N (H) H Z[i] char R Z(p) deg f (x) R[x] Page 351 Symbol Description R[x1 , x2 , , xn ] ϕα Q(x) ν(a) F (x) F (x1 , , xn ) a⪯b a∨b a∧b I O a′ dim V U ⊕V Hom(V, W ) ring of polynomials in n indeterminants evaluation homomorphism at α field of rational functions over Q Euclidean valuation of a field of rational functions in x field of rational functions in x1 , , xn a is less than b join of a and b meet of a and b largest element in a lattice smallest element in a lattice complement of a in a lattice dimension of a vector space V direct sum of vector spaces U and V set of all linear transformations from U into V dual of a vector space V smallest field containing F and α1 , , αn dimension of a field extension of E over F Galois field of order pn multiplicative group of a field F Galois group of E over F field fixed by the automorphism σi field fixed by the automorphism group G discriminant of a polynomial V∗ F (α1 , , αn ) [E : F ] GF(pn ) F∗ G(E/F ) F{σi } FG ∆2 Page 222 222 238 242 246 246 249 251 251 252 252 253 269 271 272 272 275 278 295 296 310 313 314 326 352 APPENDIX C NOTATION Index G-equivalent, 176 G-set, 175 nth root of unity, 53, 318 rsa cryptosystem, 88 Cardano, Gerolamo, 230 Carmichael numbers, 93 Cauchy’s Theorem, 189 Cauchy, Augustin-Louis, 67 Cayley table, 34 Cayley’s Theorem, 122 Cayley, Arthur, 122 Centralizer of a subgroup, 178 Characteristic of a ring, 204 Chinese Remainder Theorem for integers, 211 Cipher, 85 Ciphertext, 85 Circuit parallel, 257 series, 257 series-parallel, 258 Class equation, 178 Code bch, 301 cyclic, 295 group, 102 linear, 105 minimum distance of, 100 polynomial, 296 Commutative diagrams, 142 Commutative rings, 199 Composite integer, 22 Composition series, 170 Congruence modulo n, 13 Conjugacy classes, 178 Conjugate elements, 309 Conjugate, complex, 50 Conjugation, 176 Constructible number, 283 Correspondence Theorem for groups, 143 Abel, Niels Henrik, 317 Abelian group, 34 Adleman, L., 88 Algebraic closure, 279 Algebraic extension, 273 Algebraic number, 274 Algorithm division, 223 Euclidean, 22 Ascending chain condition, 240 Associate elements, 238 Atom, 255 Automorphism inner, 145 Basis of a lattice, 157 Bieberbach, L., 160 Binary operation, 33 Binary symmetric channel, 99 Boole, George, 259 Boolean algebra atom in a, 255 definition of, 253 finite, 255 isomorphism, 255 Boolean function, 183, 262 Burnside’s Counting Theorem, 180 Burnside, William, 38, 136, 185 Cancellation law for groups, 37 for integral domains, 203 353 354 for rings, 207 Coset leader, 112 left, 77 representative, 77 right, 77 Coset decoding, 111 Cryptanalysis, 86 Cryptosystem rsa, 88 affine, 87 definition of, 85 monoalphabetic, 86 polyalphabetic, 87 private key, 86 public key, 85 single key, 86 Cycle definition of, 63 disjoint, 63 De Morgan’s laws for Boolean algebras, 255 for sets, De Morgan, Augustus, 259 Decoding table, 112 Deligne, Pierre, 287 DeMoivre’s Theorem, 52 Derivative, 292 Determinant, Vandermonde, 299 Dickson, L E., 136 Diffie, W., 88 Direct product of groups external, 124 internal, 125 Discriminant of the cubic equation, 234 of the quadratic equation, 233 Division algorithm for integers, 20 for polynomials, 223 Division ring, 200 Domain Euclidean, 242 principal ideal, 239 unique factorization, 239 Doubling the cube, 285 Eisenstein’s Criterion, 228 Element associate, 238 identity, 34 inverse, 34 irreducible, 238 order of, 48 prime, 238 primitive, 311 transcendental, 273 Equivalence class, 12 Equivalence relation, 11 Euclidean algorithm, 22 Euclidean domain, 242 Euclidean group, 153 Euclidean inner product, 150 Euclidean valuation, 242 Euler ϕ-function, 81 Euler, Leonhard, 81, 286 Extension algebraic, 273 field, 271 finite, 276 normal, 313 radical, 318 separable, 292, 310 simple, 273 External direct product, 124 Faltings, Gerd, 287 Feit, W., 136, 185 Fermat’s factorizationalgorithm, 92 Fermat’s Little Theorem, 81 Fermat, Pierre de, 81, 286 Ferrari, Ludovico, 230 Ferro, Scipione del, 229 Field, 200 algebraically closed, 279 base, 271 extension, 271 fixed, 312 Galois, 293 of fractions, 237 of quotients, 237 splitting, 280 Finitely generated group, 165 Fior, Antonio, 230 First Isomorphism Theorem for groups, 141 for rings, 207 Fixed point set, 177 Freshman’s Dream, 292 Function bijective, Boolean, 183, 262 composition of, 355 definition of, domain of, identity, 10 injective, invertible, 10 one-to-one, onto, range of, surjective, switching, 183, 262 Fundamental Theorem of Algebra, 279, 322 of Arithmetic, 22 of Finite Abelian Groups, 166 Fundamental Theorem of Galois Theory, 314 Galois field, 293 Galois group, 308 Galois, Évariste, 37, 317 Gauss’s Lemma, 243 Gauss, Karl Friedrich, 245 Gaussian integers, 203 Generator of a cyclic subgroup, 48 Generators for a group, 165 Glide reflection, 154 Gorenstein, Daniel, 136 Greatest common divisor of two integers, 20 of two polynomials, 224 Greatest lower bound, 250 Greiss, R., 136 Grothendieck, Alexander, 287 Group p-group, 189 abelian, 34 action, 175 alternating, 67 center of, 178 circle, 53 commutative, 34 cyclic, 48 definition of, 33 dihedral, 68 Euclidean, 153 factor, 132 finite, 36 finitely generated, 165 Galois, 308 general linear, 35, 149 generators of, 165 homomorphism of, 139 infinite, 36 isomorphic, 119 isomorphism of, 119 nonabelian, 34 noncommutative, 34 of units, 35 order of, 36 orthogonal, 150 permutation, 62 point, 158 quaternion, 36 quotient, 132 simple, 133, 136 solvable, 172 space, 158 special linear, 39, 149 special orthogonal, 153 symmetric, 61 symmetry, 155 Groupp-group, 166 Gödel, Kurt, 259 Hamming distance, 100 Hamming, R., 102 Hellman, M., 88 Hilbert, David, 160, 209, 259, 287 Homomorphic image, 139 Homomorphism canonical, 141, 207 evaluation, 205, 222 kernel of a group, 140 kernel of a ring, 204 natural, 141, 207 of groups, 139 ring, 204 Ideal definition of, 205 maximal, 208 one-sided, 206 prime, 208 principal, 206 trivial, 205 two-sided, 206 Indeterminate, 219 Index of a subgroup, 78 Induction first principle of, 18 second principle of, 19 Infimum, 250 Inner product, 104 Integral domain, 200 Internal direct product, 125 356 International standard book number, 44 Irreducible element, 238 Irreducible polynomial, 225 Isometry, 154 Isomorphism of Boolean algebras, 255 of groups, 119 ring, 204 Join, 251 Jordan, C., 136 Jordan-Hölder Theorem, 171 Kernel of a group homomorphism, 140 of a ring homomorphism, 204 Key definition of, 85 private, 86 public, 85 single, 86 Klein, Felix, 38, 147, 209 Kronecker delta, 108, 151 Kronecker, Leopold, 286 Kummer, Ernst, 286 Lagrange’s Theorem, 79 Lagrange, Joseph-Louis, 37, 67, 81 Laplace, Pierre-Simon, 67 Lattice completed, 253 definition of, 251 distributive, 253 Lattice of points, 157 Lattices, Principle of Duality for, 251 Least upper bound, 250 Left regular representation, 122 Lie, Sophus, 38, 192 Linear combination, 265 Linear dependence, 266 Linear independence, 266 Linear map, 147 Linear transformation definition of, 9, 147 Lower bound, 250 Mapping, see Function Matrix distance-preserving, 151 generator, 106 inner product-preserving, 151 invertible, 148 length-preserving, 151 nonsingular, 149 null space of, 104 orthogonal, 150 parity-check, 105 similar, 12 unimodular, 157 Matrix, Vandermonde, 299 Maximal ideal, 208 Maximum-likelihood decoding, 98 Meet, 251 Minimal generator polynomial, 297 Minimal polynomial, 274 Minkowski, Hermann, 287 Monic polynomial, 220 Mordell-Weil conjecture, 287 Multiplicity of a root, 310 Noether, A Emmy, 209 Noether, Max, 209 Normal extension, 313 Normal series of a group, 170 Normal subgroup, 131 Normalizer, 191 Null space of a matrix, 104 Odd Order Theorem, 195 Orbit, 176 Orthogonal group, 150 Orthogonal matrix, 150 Orthonormal set, 151 Partial order, 249 Partially ordered set, 249 Partitions, 12 Permutation cycle structure of, 82 definition of, 9, 61 even, 66 odd, 66 Permutation group, 62 Plaintext, 85 Polynomial code, 296 content of, 243 definition of, 219 degree of, 220 error, 304 error-locator, 304 357 greatest common divisor of, 224 in n indeterminates, 222 irreducible, 225 leading coefficient of, 220 minimal, 274 minimal generator, 297 monic, 220 primitive, 243 root of, 224 separable, 310 zero of, 224 Polynomial separable, 292 Poset definition of, 249 largest element in, 252 smallest element in, 252 Power set, 249 Prime element, 238 Prime ideal, 208 Prime integer, 22 Primitive nth root of unity, 53, 318 Primitive element, 311 Primitive Element Theorem, 311 Primitive polynomial, 243 Principal ideal, 206 Principal ideal domain (pid), 239 Principal series, 170 Pseudoprime, 93 Quaternions, 36, 201 Resolvent cubic equation, 234 Rigid motion, 32, 154 Ring characteristic of, 204 commutative, 199 definition of, 199 division, 200 factor, 207 homomorphism, 204 isomorphism, 204 Noetherian, 240 quotient, 207 with identity, 199 with unity, 199 Rivest, R., 88 Ruffini, P., 317 Russell, Bertrand, 259 Scalar product, 263 Second Isomorphism Theorem for groups, 142 for rings, 207 Shamir, A., 88 Shannon, C., 102 Simple extension, 273 Simple group, 133 Simple root, 310 Solvability by radicals, 318 Spanning set, 265 Splitting field, 280 Squaring the circle is impossible, 286 Standard decoding, 111 Subgroup p-subgroup, 189 centralizer, 178 commutator, 193 cyclic, 48 definition of, 38 index of, 78 isotropy, 177 normal, 131 normalizer of, 191 proper, 38 stabilizer, 177 Sylowp-subgroup, 190 translation, 158 trivial, 38 Subnormal series of a group, 169 Subring, 202 Supremum, 250 Switch closed, 257 definition of, 257 open, 257 Switching function, 183, 262 Sylow p-subgroup, 190 Sylow, Ludvig, 192 Syndrome of a code, 111, 304 Tartaglia, 230 Third Isomorphism Theorem for groups, 143 for rings, 207 Thompson, J., 136, 185 Transcendental element, 273 Transcendental number, 274 Transposition, 65 Trisection of an angle, 286 Unique factorization domain (ufd), 239 Unit, 200, 238 358 Universal Product Code, 43 Upper bound, 250 Vandermonde determinant, 299 Vandermonde matrix, 299 Vector space basis of, 266 definition of, 263 dimension of, 267 subspace of, 264 Weight of a codeword, 100 Weil, André, 287 Well-defined map, Well-ordered set, 19 Whitehead, Alfred North, 259 Zero multiplicity of, 310 of a polynomial, 224 Zero divisor, 200 Colophon This book was authored and produced with PreTeXt ... object To show that it is unique, assume that there are two such objects, say r and s, and then show that r = s • Sometimes it is easier to prove the contrapositive of a statement Proving the statement... introductory abstract algebra course is that it should be written to convince one’s peers, whether those peers be other students or other readers of the text Let us examine different types of statements... section presenting hints and solutions to many of the exercises appears at the end of the text Often in the solutions a proof is only sketched, and it is up to the student to provide the details