A book of abstract algebra

Cover......Page 1
A BOOK OF ABSTRACT ALGEBRA......Page 4
QA162.P56 512\'.02......Page 5
Dedication......Page 6
CONTENTS......Page 8
PREFACE......Page 14
ACKNOWLEDGMENTS......Page 16
CHAPTER ONE: WHY ABSTRACT ALGEBRA?......Page 18
ORIGINS......Page 20
The Algebra of Matrices......Page 24
Boolean Algebra......Page 27
Algebraic Structures......Page 28
AXIOMS AND MEN......Page 29
THE AXIOMATICS OF ALGEBRA......Page 31
ABSTRACTION REVISITED......Page 33
CHAPTER TWO: OPERATIONS......Page 37
B. Properties of Operations......Page 40
C. Operations on a Two-Element Set......Page 42
CHAPTER THREE: THE DEFINITiON OF GROUPS......Page 43
A. Examples of Abelian Groups......Page 47
C. Groups of Subsets of a Set......Page 48
D. A Checkerboard Game......Page 49
K A Coin Game......Page 50
F. Groups in Binary Codes......Page 51
CHAPTER FOUR: ELEMENTARY PROPERTiES OF GROUPS......Page 53
A. Solving Equations in Groups......Page 56
D. Group Elements and Their Inverses......Page 57
E. Counting Elements and Their Inverses......Page 58
F. Constructing Small Groups......Page 59
H. Powers and Roots of Group Elements......Page 60
CHAPTER FIVE: SUBGROUPS......Page 62
A. Recognizing Subgroups......Page 66
C. Subgroups of Abelian Groups......Page 67
E. Generators of Groups......Page 68
G. Cayley Diagrams......Page 69
CHAPTER SIX: FUNCTiONS......Page 71
A. Examples of Injective and Surjective Functions......Page 77
D. Composite Functions......Page 78
E. Inverses of Functions......Page 79
G. Some General Properties of Functions......Page 80
CHAPTER SEVEN: GROUPS OF PERMUTATIONS......Page 81
A. Computing Elements of S5......Page 87
B. Examples of Groups of Permutations......Page 88
F. Symmetries of Geometric Figures......Page 89
G. Symmetries of Polynomials......Page 90
H. Properties of Permutations of a Set A......Page 91
CHAPTER EIGHT: PERMUTATIONS OF A FINITE SET......Page 92
A. Practice in Multiplying and Factoring Permutations......Page 98
B. Powers of Permutations......Page 99
E. Conjugate Cycles......Page 100
G. Even/Odd Permutations in Subgroups of Sn......Page 101
H. Generators of An and Sn......Page 102
CHAPTER NINE: 1SOMORPHISM......Page 103
B. Elements Which Correspond under an Isomorphism......Page 111
D. Separating Groups into Isomorphism Classes......Page 112
F. Isomorphism of Groups Given by Generators and Defining Equations......Page 113
I. Group Automorphisms......Page 114
J. Regular Representation of Groups......Page 115
CHAPTER TEN: ORDER OF GROUP ELEMENTS......Page 117
A. Laws of Exponents......Page 121
C. Elementary Properties of Order......Page 122
F. Orders of Powers of Elements......Page 123
H. Relationship between the Order of a and the Order of any kth Root of a......Page 124
CHAPTER ELEVEN: CYCLIC GROUPS......Page 126
C. Generators of Cyclic Groups......Page 130
E. Direct Products of Cyclic Groups......Page 131
F. kth Roots of Elements in a Cyclic Group......Page 132
CHAPTER TWELVE: PARTITIONS AND EQUIVALENCE RELATIONS......Page 133
B. Examples of Equivalence Relations......Page 138
E. General Properties of Equivalence Relations and Partitions......Page 139
CHAPTER THIRTEEN: COUNTING COSETS......Page 140
B. Examples of Cosets in Infinite groups......Page 144
E. Elementary Properties of Cosets......Page 145
G. Survey of All 10-Element Groups......Page 146
 I. Conjugate Elements......Page 147
 J. Group Acting on a Set......Page 148
CHAPTER FOURTEEN: HOMOMORPHISMS......Page 149
A. Examples of Homomorphisms of Finite Groups......Page 154
C. Elementary Properties of Homomorphisms......Page 156
F. Homomorphism and the Order of Elements......Page 157
 I. Conjugate Subgroups......Page 158
CHAPTER FIFTEEN: QUOTIENT GROUPS......Page 160
A. Examples of Finite Quotient Groups......Page 165
C. Relating Properties of H to Properties of G/H......Page 166
F. Quotient of a Group by its Center......Page 167
G. Using the Class Equation to Determine the Size of the Center......Page 168
H. Induction on I G I An Example......Page 169
CHAPTER SIXTEEN: THE FUNDAMENTAL HOMOMORPHISM THEOREM......Page 170
A. Examples of the FHT Applied to Finite Groups......Page 173
D. Group of Inner Automorphisms of a Group G......Page 174
F. The First Isomorphism Theorem......Page 175
H. Quotient Groups Isomorphic to the Circle Group......Page 176
K. Cauchy \'s Theorem......Page 177
 M. p-Sylow Subgroups......Page 178
N. Sylow\'s Theorem.......Page 179
P. Decomposition of a Finite Abelian Group into p-Groups......Page 180
Q. Basis Theorem for Finite Abelian Groups......Page 181
CHAPTER SEVENTEEN: RINGS: DEFINiTIONS AND ELEMENTARY PROPERTIES......Page 182
C. Ring of 2 x 2 Matrices......Page 188
E. Ring of Quaternions......Page 189
F. Ring of Endomorphisms......Page 190
I. Properties of Invertible Elements......Page 191
L. The Binomial Formula......Page 192
M. Nilpotent and Unipotent Elements......Page 193
CHAPTER EIGHTEEN: IDEALS AND HOMOMORPHISMS......Page 195
A. Examples of Subrings......Page 199
C. Elementary Properties of Subrings......Page 200
F. Elementary Properties of Homomorphisms......Page 201
H. Further Properties of Ideals......Page 202
J. A Ring of Endomorphisms......Page 203
CHAPTER NINETEEN: QUOTIENT RINGS......Page 204
A. Examples of Quotient Rings......Page 209
C. Quotient Rings and Homomorphic Images in 39R)......Page 210
E. Properties of Quotient Rings A/J in Relation to Properties of J......Page 211
H. Zn as a Homomorphic Image of Z......Page 212
CHAPTER TWENTY: INTEGRAL DOMAiNS......Page 214
OPTIONAL......Page 217
B. Characteristic of a Finite Integral Domain......Page 219
E. Further Properties of the Characteristic of an Integral Domain......Page 220
F. Finite Fields......Page 221
CHAPTER TWENTY-ONE: THE INTEGERS......Page 222
C. Uses of Induction......Page 228
E. Absolute Values......Page 229
H. Principle of Strong Induction......Page 230
CHAPTER TWENTY-TWO: FACTORING INTO PRIMES......Page 232
B. Properties of the gcd......Page 238
F. Least Common Multiples......Page 239
H. The gcd and the 1cm as Operations on /.......Page 240
CHAPTER TWENTY-THREE: ELEMENTS OF NUMBER THEORY......Page 241
OPTIONAL......Page 245
A. Solving Single Congruences......Page 249
C. Elementary Properties of Congruence......Page 250
FL Consequences of Fermat\'s Theorem......Page 251
G. Wilson\'s Theorem, and Some Consequences......Page 252
H. Quadratic Residues......Page 253
I. Primitive Roots......Page 255
CHAPTER TWENTY-FOUR: RINGS OF POLYNOMIALS......Page 256
B. Problems Involving Concepts and Definitions......Page 263
D. Domains A(x) where A Has Finite Characteristic......Page 264
F. Homomorphisms of Domains of Polynomials......Page 265
I. Fields of Polynomial Quotients......Page 266
J. Division Algorithm: Uniqueness of Quotient and Remainder......Page 267
CHAPTER TWENTY-FIVE: FACTORING POLYNOMIALS......Page 268
B. Short Questions Relating to Irreducible Polynomials......Page 272
F. A Method for Computing the gcd......Page 273
G. An Automorphism of 194......Page 274
CHAPTER TWENTY-SIX: SUBSTITUTION IN POLYNOMIALS......Page 275
POLYNOMIALS OVER ZAND Q......Page 278
POLYNOMIALS OVER  R AND C......Page 281
B. Finding Roots of Polynomials over Q......Page 282
D. Irreducible Polynomials in Q(x) by Eisenstein \'s Criterion\r(and Variations on the Theme)......Page 283
F. Mapping onto Zn to Determine Irreducibility over Q......Page 284
H. Polynomial Functions over a Finite Field......Page 285
I. Polynomial Interpolation......Page 286
CHAPTER TWENTY-SEVEN: EXTENSIONS OF FiELDS......Page 287
A. Recognizing Algebraic Elements......Page 293
C. The Structure of Fields F(x)/......Page 294
E. Simple Extensions......Page 295
G. Questions Relating to Transcendental Elements......Page 296
J. Multiple Roots......Page 297
CHAPTER TWENTY-EIGHT: VECTOR SPACES......Page 299
B. Examples of Subspaces......Page 306
E. Properties of Linear Transformations......Page 307
G. Sums of Vector Spaces......Page 308
CHAPTER TWENTY-NINE: DEGREES OF FIELD EXTENSIONS......Page 309
A. Examples of Finite Extensions......Page 314
D. Degrees of Extensions (Applications of Theorem 2)......Page 315
F. Further Properties of Degrees of Extensions......Page 316
G. Fields of Algebraic Elements: Algebraic Numbers......Page 317
CHAPTER THIRTY: RULER AND COMPASS......Page 318
A. Constructible Numbers......Page 324
D. Constructible Polygons......Page 325
F. A Nonconstructible Polygon......Page 326
G. Further Properties of Constructible Numbers and Figures......Page 327
CHAPTER THIRTY-ONE: GALOIS THEORY: PREAMBLE......Page 328
B. Examples of Root Fields over Zp......Page 334
D. Reducing Iterated Extensions to Simple Extensions......Page 335
F. Separable and Inseparable Polynomials......Page 336
G. Multiple Roots over Infinite Fields of Nonzero Characteristic......Page 337
J. Extending Isomorphisms......Page 338
K. Normal Extensions......Page 339
CHAPTER THIRTY-TWO: GALOIS THEORY: THE HEART OF THE MATTER......Page 340
A. Computing a Galois Group......Page 347
D. A Galois Group Equal to D4......Page 348
F. A Galois Group Isomorphic to S5......Page 349
I. Further Questions Relating to Galois Groups......Page 350
J. Normal Extensions and Normal Subgroups......Page 351
CHAPTER THIRTY-THREE: SOLVING EQUATIONS BY RADICALS......Page 352
A. Finding Radical Extensions......Page 359
C. pth Roots of Elements in a Field......Page 360
E. If Gal(K: F) Is Solvable, K Is a Radical Extension of F......Page 361
INDEX......Page 364