Classical abstract algebra:
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| Main Author: | |
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| Format: | Book |
| Language: | English |
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New York u.a.
Harper & Row
1990
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| Online Access: | Inhaltsverzeichnis |
| Physical Description: | XXIX, 524 S. graph. Darst. |
| ISBN: | 0060416017 |
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MARC
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| 100 | 1 | |a Dean, Richard A. |e Verfasser |4 aut | |
| 245 | 1 | 0 | |a Classical abstract algebra |c Richard A. Dean |
| 264 | 1 | |a New York u.a. |b Harper & Row |c 1990 | |
| 300 | |a XXIX, 524 S. |b graph. Darst. | ||
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| adam_text | CLASSICAL
ABSTRACT
ALGEBRA
Richard A Dean
California Institute of Technology
HARPER amp; ROW, PUBLISHERS, New York
Grand Rapids, Philadelphia, St Louis, San Francisco,
1817 London, Singapore, Sydney, Tokyo
Contents
Preface xxlii
PART ONE BASIC CONCEPTS 1
Chapter 1 INTEGERS AND SETS 3
1 1 Sets 4
111 Equality of Two Sets 4
112 Subset 5
113 Properties of the (c) Relation 6
1 2 Basic Properties of the Integers 6
121 Axioms of Addition and Multiplication for the Integers 6
122 Axioms of the Order Relation on the Integers 8
123 Discrete Nature of Z 10
124 Division Algorithm for I 10
125 Mathematical Induction 11
Exercise Set 1 2 14
1 3 Integer Arithmetic 15
131 Division in the Integers 15
132 Division Properties 15
133 Greatest Common Divisor (gcd) 16
134 The Division Algorithm and gcd 17
Vi CONTENTS
135 Existence of the gcd 18
136 The Euclidean Algorithm to Find gcd (a, b) 19
137 Greatest Common Divisor Properties 21
138 Prime Integer 22
139 Alternative Definition of a Prime 22
1 3 10 Unique Factorization 23
1 3 11 The Sieve of Eratosthenes 25
1 3 12 Infinitude of Primes 26
1 3 13 Ideal 27
1 3 14 Elementary Ideal Properties 27
1 3 15 Characterization of the Ideals in Z 28
Exercise Set 1 3 28
1 4 More on Sets 30
141 Number of Elements in a Power Set 30
142 Set Operations 31
143 Disjoint Sets 31
144 Properties of Set Operations 32
145 Partition 33
146 Diagrams 33
147 Ordered Pair 34
148 Cartesian Product 35
149 Function 35
1 4 10 Binary Operations and Tables 36
1 4 11 Composition of Functions Is Associative 37
Exercise Set 1 4 37
Chapter 2 RELATIONS AND LATTICES 40
2 1 Equivalence Relations and Congruence in Z 40
211 Congruence Modulo 10 on Z 41
212 Equivalence Relation 42
213 Equivalence Classes 43
214 Equivalence Relations and Partitions 43
215 Congruence Modulo n on Z 44
216 Equality Properties of Congruence 44
217 Changing Moduli of Congruences 45
218 Solution of Linear Congruences 45
219 Operations on Congruence Classes 48
2 1 10 Operations Are Well Defined 49
2 1 11 Arithmetic Properties of Zn 49
2 1 12 Congruence Classes and gcds 50
2 1 13 Euler s ^-Function 51
CONTENTS Vii
2 1 14 A Multiplicative Property 51
Exercise Set 2 1 52
2 2 Partially Ordered Sets and Lattices 54
221 Partially Ordered Set (POS) 54
222 Chain 56
223 Cover 56
224 Maximal and Minimal Elements 56
225 Bounds 57
226 Maximal Principle 57
227 Least Upper Bound (lub) and Greatest Lower
Bound (gib) 58
228 lubs and gibs in (Z +, | 58
229 Dual Partial Ordering 58
2 2 10 Lattice 58
2 2 11 Examples of Lattices 59
2 2 12 Complete Lattice 60
2 2 13 Sufficient Condition for a Complete Lattice 60
2 2 14 Lattice Properties 62
2 2 15 Lattice of Equivalence Relations 63
Exercise Set 2 2 64
Chapter 3 INTRODUCTION TO GROUPS 66
3 1 Group Axioms 67
311 Group 67
312 Abelion Group 67
313 Order of a Group 68
314 Examples of Groups 68
315 Idempotent Element 72
316 Idempotence Implies Identity 72
317 Uniqueness of the Right Identity Element 72
318A Right Inverse Is a Left Inverse 73
319A Right Identity Is a Left Identity 73
3 1 10 Uniqueness of Each Inverse 73
3 1 11 Solution of Group Equations 74
3 1 12 Latin Square 75
3 1 13 Latin Square Property of Group Tables 75
3 1 14 Elementary Group Computations 76
3 1 15 Group Isomorphism 77
3 1 16 Preservation of Identity and Inverse under Isomorphism 78
Exercise Set 3 1 79
Viii CONTENTS
3 2 Subgroups 81
321 Symmetries of the Square 81
322 Subgroup 83
323 Criteria for Subgroups 83
324 Cyclic Groups and the Order of an Element 84
325 Subgroups of Cyclic Groups 85
326 Properties of the Order of an Element 88
327 Lattice of Subgroups 88
328A Characterization of Subgroup Join 89
329 The Subgroups of D8 90
3 2 10 Subgroup Generated by a Set 92
Exercise Set 3 2 93
3 3 Dihedral Groups 95
331 The Group of Symmetries of the Regular n-gon 95
332 Dihedral Group of Order 2n (n 2) 96
333 Order of a Dihedral Group 97
Exercise Set 3 3 98
3 4 Counting Theorems 98
341 Theorem of Lagrange 99
342 Coset 100
343 Examples of Cosets 100
344 Coset Partitions 102
345 Index of a Subgroup 102
346 Second Version of the Theorem of Lagrange 103
347 Applications and Extensions of Lagrange s Theorem 103
348 Group Complex of Two Subgroups 104
349 The Number of Elements in a Complex 104
3 4 10 Groups of Order 6 105
3 4 11 A Theorem of Cauchy 106
3 4 12 Cyclic Shifts 107
3 4 13 Groups of Order 15 108
Exercise Set 3 4 109
Chapter 4 GROUP HOMOMORPHISMS 111
4 1 Regular Tetrahedrons and the Alternating Group j^4 111
411 Physical Model 111
412 The Group of Rotations of a Regular Tetrahedron 112
413 The Alternating Group sf4 114
414 Subgroup Lattice of sf4 114
Exercise Set 4 1 115
CONTENTS JX
4 2 Group Homomorphisms 116
421A Group Model of s?4 116
422 Homomorphism 117
423 Isomorphism, Endomorphism, and Automorphism 117
424A Homomorphism of the Four-group 118
425A Homomorphism of sf4 118
426 Elementary Properties of Homomorphisms 118
427 Three Equivalent Properties 120
428 Normal Subgroup 121
429 Examples of Normal Subgroups 121
4 2 10 A Normal Subgroup Is a Kernel 122
4 2 11 Factor Group 123
4 2 12 The Factor Group st4/ V 124
4 2 13 Subgroup Properties of Homomorphisms 125
Exercise Set 4 2 127
4 3 Normal Subgroups 128
431 The Normal Subgroups of s/4 128
432 The Quaternion Group 129
433 Sufficient Condition for a Complex AB to Be a
Subgroup 131
434 Lattice Properties of Normal Subgroups 131
435 The Diamond Isomorphism 132
436A Normality Criterion 133
Exercise Set 4 3 134
Chapter 5 DIRECT PRODUCTS 137
5 1 Groups of Low Order 137
511 Groups of Order p and 2p 137
512 An Abelian Group of Order 8 139
513 The Elementary Abelian Group of Order 8 140
514 The Groups of Order 8 142
515 The Groups of Order 9 144
Exercise Set 5 1 145
5 2 Direct Products of Groups 146
521 Direct Product of Two Groups 146
522 Properties of the Direct Product 146
523 Examples of Direct Products 147
524A Criterion for a Group to Be a Direct Product 148
525 The Groups of Order 9 149
526A Direct Product Decomposition of Z*m 149
527A Direct Product of Several Groups 151
CONTENTS
528A Criterion for a Group to Be the Direct Product
of Several Subgroups 151
Exercise Set 5 2 152
5 3 Finite Abelian Groups 152
531 Structure of Finite Abelian Groups 153
532 Expressing an Element as the Product of
Commuting Elements 154
533 An Element of Order the km of Two Orders 154
534 Exponent of an Abelian Group 155
535 Exponent of a Group 155
536 Criterion for a Group to Be Cyclic 156
Exercise Set 5 3 156
Chapters PERMUTATION GROUPS 158
6 1 Transpositions and the Alternating Groups 158
611 Permutation 158
612 Full Symmetric Group S n 159
613 Permutations Fixing a Subset 160
614 Transposition 160
615 The Transpositions in S4 160
616A Permutation Is a Product of Transpositions 161
617 Even and Odd Permutations 161
618 The Identity Is not an Odd Permutation 162
619 No Permutation Is Both Even and Odd 163
6 1 10 The Alternating Group s/n 163
6 1 11 Two Properties of s/n 163
Exercise Set 6 1 164
6 2 n-Cycles and the Simplicity of s/a (n 5) 164
621 n-Cycle 165
622 Cycles in S4 165
623 Properties of n-Cycles 166
624A Permutation Is the Product of Disjoint Cycles 166
625 Cyclic Notation for Permutations 168
626 Generators for the Alternating Group 168
627 Conjugates of Cycles 169
628 Invariance of Cycle Structure under Conjugation 169
629 Conjugacy of k-cycles in Sn 169
6 2 10 A Normal Subgroup of s/n Contains a Permutation Fixing
an Element 170
6 2 11 Sufficient Condition for Normality in s0n to Imply Equality
toj*» 171
CONTENTS XI
6 2 12 Alternating Group s/n Simple if n 5 172
Exercise Set 6 2 172
6 3 Representations of Groups by Permutations 173
631 The Right Regular Representation 173
632 Representation by Permutation of Cosets 174
633 Criterion for a Proper Normal Subgroup 175
634 Groups of Order pm, p m 176
635A Group of Order 24 Has a Proper Normal Subgroup 176
636 Groups of Order p2 176
Exercise Set 6 3 176
Chapter 7 INTRODUCTION TO RINGS 178
7 1 Ring Axioms and Elementary Properties 178
711 Ring 178
712 Familiar Rings 179
713 Special Classes of Rings 182
714 Properties of Rings 184
715 Finite Integral Domains 184
716 Cancellation 185
717 Units and Associates 185
718 The Group of Units 185
719 Units in Familiar Rings 186
Exercise Set 7 1 186
7 2 Subrings, Homomorphisms, and Factor Rings 187
721 Subring 188
722 Lattice of Subrings 188
723A Form for Elements in the Join of Subrings 188
724 Ring Homomorphisms 189
725 Ideal 190
726 Independence of Residue Class Representatives 191
727 Factor Ring 191
728 Z„ as a Factor Ring 191
729 The Subring (I) 192
7 2 10 Characteristic of a Ring 192
7 2 11 Congruence on a Ring 193
7 2 12 Ideals and Congruence Relations 193
7 2 13 Lattice Properties under Homomorphisms 193
7 2 14 Direct Sum of Rings 194
xii CONTENTS
7 2 75 Criteria for a Ring to Be a Direct Sum 195
Exercise Set 7 2 196
Chapter 8 INTEGRAL DOMAINS 198
8 1 Arithmetic in an Integral Domain 198
811 Division in Commutative Rings 198
812 Properties of Division 199
813 Irreducible and Prime Elements 199
814 Primes Are Irreducibles 199
815 An Irreducible That Is Not Prime 200
816 Greatest Common Divisor (gcd) and Least Common
Multiple (km) 201
817 Uniqueness of gcd and km to Within Units 201
818 Two Elements Having a gcd but Not an km 201
819A Relation between gcd and km 202
Exercise Set 8 1 203
8 2 Ideals in Commutative Rings 203
821 The Join of Ideals and the Diamond Lemma for Rings 203
822 Construction of an Ideal 204
823 Principal Ideal 205
824 Prime Ideal 205
825 Criterion for a Principal Ideal to Be Prime 205
826 Criterion for a Factor Ring to Be an Integral Domain 206
827 Zp Is a Field 206
828 Maximal Ideal 206
829 Criterion for a Factor Ring to Be a Field 207
Exercise Set 8 2 207
8 3 Introduction to Fields 208
831 Characteristic of a Field 209
832 Subfield 209
833 Field Extensions and Embeddings 209
834A Field with Four Elements 209
835 Lattice of Subfields 211
836 Construction of the Field of Fractions 211
837 Prime Subfield 212
838 Characterization of the Prime Subfield of a Field 212
839 Fixed Field of an Automorphism 213
8 3 10 Ring Automorphisms Form a Group 213
Exercise Set 8 3 214
CONTENTS xiii
Chapter 9 POLYNOMIALS 216
9 1 Polynomial Rings 217
911 Polynomials over a Ring 217
912 Ring Properties of Polynomials 218
913 Subring of Scalars 219
914 The Indeterminate x and the Ring R[x] 219
915 If R is an Integral Domain So Is R[x] 220
916 Substitution for x 221
917 Zero of a Polynomial 222
918 Polynomial Function 222
919A Function That Is Not a Polynomial Function 223
9 1 10 Extensions of Homomorphisms and Zeros 224
9 1 11 Division Algorithm in R[x] 225
9 1 12 Remainder Theorem 226
9 1 13 Factor Theorem 227
9 1 14 Polynomial Ring in Several Indeterminates 227
9 1 15 Representation for Elements in R[xv , xn] 227
Exercise Set 9 1 229
9 2 Polynomials over a Field 230
921 Arithmetic in F[x] 230
922 Unique Factorization in F[x] 233
923 F[x] Is a Principal Ideal Domain 233
924A Sufficient Condition for f(x) to Be Reducible 234
925 Reducibility Test for Quadratics and Cubics 234
926 Preservation of the gcd 234
927 Test for a Common Zero 235
928 Number of Zeros of a Polynomial 235
929 Lagrange Interpolation Formula 236
9 2 10 Multiple Zero 237
9 2 11 Formal Derivative of a Polynomial 238
9 2 12 Algebraic Properties of Derivatives 238
9 2 13 Multiple Zero Test 238
9 2 14 Sufficient Condition for Simple Zeros 239
9 2 15 Existence of Zeros 240
9 2 16 Kronecker s Extension Theorem 240
9 2 17 The Field I3[x]/(x2 + 1 242
Exercise Set 9 2 242
9 3 Euclidean Domains and Principal Ideal Domains 243
931 Euclidean Domain 244
932 Elementary Properties of Euclidean Domains 244
xiv CONTENTS
933A Principal Ideal Domain (PID) 245
934 Properties of a PID 245
935 Unique Factorization in a PID 247
936 Euclidean Algorithm in Euclidean Domains 248
Exercise Set 9 3 249
9 4 Z[x] and Other Unique Factorization Domains 249
941 I x] is Not a PID 249
942 Unique Factorization Domains 250
943 Arithmetic in UFDs 251
944 Test for Fractional Zeros in a Field of Fractions 252
945 Primitive Polynomial 252
946 Form for Polynomials in D[x] 253
947 Lemma of Gauss 253
948 Factorization in D[x] 254
949 Factorization in Q[x] 255
9 4 10 Irreducibles Are Primes in D[x] 255
9 4 11 D[x] Is a UFD if D Is a UFD 256
9 4 12 Eisenstein s Irreducibility Criterion 256
9 4 13 Application of Eisenstein s Irreducibility Criterion 257
Exercise Set 9 4 258
Chapter 10 FIELD EXTENSIONS 260
10 1 Vector Spaces 260
10 1 1 Vector Space over a Field 261
10 1 2 Vector Space of n-tuples over F 261
10 13A Field Extension as a Vector Space 262
10 1 4 Subspaces 263
10 1 5 Span of a Subset of Vectors 263
10 1 6 Lattice of Subspaces 263
10 1 7 Direct Sum of Vector Spaces 264
10 1 8 Dependence and Independence of a Set of Vectors 264
10 1 9 Basis 264
10 1 10 Examples of Bases 265
10 1 11 Existence of a Basis 265
10 1 12 Steinitz Replacement Theorem 267
10 1 13 Dimension 267
10 1 14 The Number of Elements in a Finite Field 269
10 1 15 Isomorphism of Vector Spaces of Equal Dimension 269
10 1 16 Dimension Formula 269
10 1 17 Systems of Linear Equations 270
10 1 18 Solutions of a System of Linear Equations 271
CONTENTS XV
10 1 19 Matrix and Transposed Matrix 273
10 1 20 Row Rank Equals Column Rank 274
10 1 21 Rank of a Matrix 276
10 1 22 Singular and Nonsingular Matrices 276
Exercise Set 10 1 276
10 2 Classifications of Field Extensions 278
10 2 1 Degree of E over F 278
10 2 2 An Infinite Extension 278
10 23A Finite Extension 279
10 2 4 Algebraic and Transcendental Elements 281
10 2 5 The Subfield F(a) 281
10 2 6 The Degree of [F(a):F] 281
10 2 7 Algebraic and Transcendental Extensions 282
10 2 8 Finite Degree Implies Algebraic Extension 282
10 2 9 Extensions of Degree 2 283
10 2 10 Simple Extensions 283
10 2 11 Extensions Generated by a Set of Elements 284
10 2 12 The Fields Q(j2), Q( /J1 and Qfi/2, {3) 284
10 2 13 Successive Adjunction 284
10 2 14 Multiplication of Degrees 285
10 2 15 Algebraic Elements Form a Field 286
10 2 16 Transitivity of Algebraic Extensions 287
Exercise Set 10 2 287
10 3 Splitting Fields 288
10 3 1 Splitting Field 288
10 3 2 The Splitting Field of (x2 - 2)(x2 - 3) over Q 289
10 3 3 Existence of a Splitting Field 289
10 3 4 Extension of an Isomorphism to an Isomorphism of
Splitting Fields 290
10 3 5 Uniqueness of the Splitting Field of f(x) over F 291
10 3 6 Existence of Automorphisms of a Splitting Field 292
10 3 7 The Splitting Field of x3 - 2 over Q 292
Exercise Set 10 3 295
10 4 Finite Fields 295
10 4 1 Finite Multiplicative Subgroups of a Field 296
10 4 2 Existence and Uniqueness of Finite Fields 296
10 4 3 Automorphisms of a Finite Field 297
10 4 4 Construction of GF(p) 298
10 45A Divisibility Property in Z 298
• 10 4 6 Subfields of GF(p) 299
xvi CONTENTS
10 4 7 Factorization in Zp[x] 299
10 4 8 Special Cases of Wp(n) 300
10 4 9 The Number of Monic Polynomials Irreducible
over Zp 301
10 4 10 Mobius n-Function 302
10 4 11 Automorphisms of GF(p) 302
Table of Monic Irreducible Polynomials of Low-Degree
Over lp (p = 2, 3, and 5) 303
Exercise Set 10 4 303
PART TWO SELECTED TOPICS 308
Chapter 11 TOPICS IN GROUPS 309
11 1 Conjugacy and the Class Equation of a Group 309
11 1 1 Conjugacy 309
11 1 2 Some Conjugate Sets in s/4 310
11 1 3 Properties of Conjugacy 310
11 1 4 The Class Equation of a Group 311
11 1 5 Cauchy s Theorem for Abelian Groups 312
11 1 6 Cauchy s Theorem for Finite Groups 312
11 1 7 p-Group 313
Exercise Set 11 1 313
11 2 The Sylow Theorem 313
11 2 1 The Sylow Theorem 314
11 2 2 The Groups of Order 12 315
Exercise Set 11 2 318
11 3 Semidirect Products 318
11 3 1 Internal Semidirect Product (Split Extension) 319
11 3 2 Familiar Semidirect Products 319
11 3 3 Properties of Split Extensions 319
11 3 4 External Semidirect Product of A by B 321
11 3 5 Properties of the Semidirect Product 321
11 3 6 Semidirect Product of Two Cyclic Groups 322
11 3 7 The Group (B, A: Bm =A= 1, AB = BAk) 324
Exercise Set 11 3 325
11 4 Finite Abelian Groups 325
11 4 1 The Join of Normal Subgroups and Their Direct
Product 325
11 4 2 Subgroups in the Products of Cyclic Groups 326
CONTENTS XVii
11 4 3 Existence of a Direct Factor of an Abelian Group 327
11 4 4 Fundamental Theorem for Finite Abelian Groups 329
11 4 5 When Is Z* Cyclic? 330
11 4 6 Homomorphisms of Z* 331
11 4 7 Two Related Congruences 331
11 4 8 Lifting Orders from Modp to Modp + 333
Exercise Set 11 4 335
11 5 A Simple Group of Order 168 335
11 5 1 The Order of SL(2, 7) 336
11 5 2 The Center (Z) ofSL(2,7) 336
11 5 3 Characteristic Polynomial of a Matrix 336
11 5 4 Conjugate Elements Have Equal Characteristic
Equations 337
11 5 5 PSL(2,7) Is a Simple Group of Order 168 337
Exercise Set 11 5 341
Chapter 12 TOPICS IN FIELDS 343
12 1 Compass and Straightedge Constructions 343
12 1 1 Points, Lines, Circles, and Constructive Numbers 344
12 1 2 Two Familiar Constructions 345
12 1 3 Rational Numbers Are Constructible from {0,1} 345
12 1 4 Field of Constructible Numbers 346
12 1 5 Constructions and Subfields 346
12 1 6 Necessary Condition for Constructibility 349
12 1 7 If w is Constructible from Q then [Q(a): Q] Is a
Power of 2 349
12 1 8 Sufficient Condition for Constructibility from {0,1} 349
12 1 9 Cube Root of 2 and Angle Trisection 350
Exercise Set 12 1 351
12 2 Roots of Unity and Cyclotomic Extensions 351
12 2 1 Roots of Unity 351
12 2 2 The Group of Roots of Unity 352
12 2 3 Primitive Root of Unity 353
12 2 4 The nth Cyclotomic Polynomial 353
12 2 5 Determination of a Cyclotomic Polynomial 353
12 2 6 Special Cases of QJx) 355
12 2 7 Wedderburn s Theorem on Finite Division Rings 356
12 2 8 Irreducibility of Cyclotomic Polynomials 357
12 2 9 Constructible Regular Polygons 359
12 2 10 Automorphisms of Cyclotomic Extensions 361
xviii CONTENTS
12 2 11 Automorphism Group ofx — a 362
Exercise Set 12 2 363
12 3 The Gaussian Integers 364
12 3 1 The Gaussian Integers Z[i] 364
12 3 2 The Units of Z[i] 365
12 3 3 Proximity of Gaussian Integers 366
12 3 4 Division Algorithm in the Gaussian Integers 366
12 3 5 The Gaussian Integers Are Euclidean 367
12 3 6 Properties of Z* 367
12 3 7 Primes That Are the Sum of Two Squares 368
12 3 8 Gaussian Primes and Their Norms 369
12 3 9 The Gaussian Primes 371
Exercise Set 12 3 371
12 4 Two Special Results of Finite Extensions 372
12 4 1 Existence of a Primitive Element 372
12 4 2 Simple Extensions 374
12 4 3 All or None Property of Splitting Fields 374
Exercise Set 12 4 375
Chapter 13 GALOIS THEORY FOR FIELDS
OF CHARACTERISTIC ZERO
13 1 The Fundamental Theorem of Galois Theory 376
13 1 1 The Group of an Extension and a Polynomial 377
13 1 2 The Fixed Field of a Group of Automorphisms 377
13 1 3 Zeros of Polynomials and Automorphisms of a Field 378
13 1 4 Conjugate Elements and Conjugate Fields 379
13 1 5 Normal Extension 380
13 1 6 Normal Extension and Galois Group 380
13 1 7 The Fundamental Theorem of Galois Theory 381
13 1 8 The Group of x4 - 2 over Q 384
Exercise Set 13 1 386
13 2 The Fundamental Theorem of Algebra 387
13 2 1 The Fundamental Theorem of AIgebra 387
13 22A Real Polynomial of Odd Degree Has a Real Zero 388
13 2 3 Quadratic Polynomials in C[x] Have Zeros in C 388
13 2 4 Every Real Polynomial Has a Complex Zero 389
13 2 5 Every Polynomial over C Splits in C 389
CONTENTS Xix
13 3 The Discriminant 390
13 3 1 The Discriminant of a Polynomial 391
13 3 2 Role of the Square Root of the Discriminant 391
13 3 3 Computation of the Discriminant from the Derivative 392
13 3 4 The Discriminant of a Quadratic and a Cubic 393
Exercise Set 13 3 394
Chapter 14 SOLUTIONS IN RADICALS 395
14 1 Solution in Radicals: A Necessary Condition 396
14 1 1 Root Tower 396
14 1 2 Solution in Terms of Radicals 397
14 1 3 The Necessary Condition 397
14 1 4 Solvable Group 398
14 1 5 Some Solvable Groups 398
14 2 Constructing Solutions in Terms of Radicals 399
14 2 1 Lagrange Resolvent 399
14 2 2 An Element Fixed by a 400
14 2 3 Cyclic Extensions 400
14 2 4 Solving the Cubic 402
Exercise Set 14 2 406
14 3 A Sufficient Condition for Solution in Radicals 407
14 3 1 Join of Normal Extensions 407
14 3 2 Solvability of the Galois Group Implies Solvability in Terms
of Radicals 408
14 4 Solvable Groups 409
14 4 1 Solvability Is Hereditary 409
14 4 2 Subnormal Chains and Composition Series 411
14 4 3 Equivalent Conditions for Solvability 412
14 4 4 Derived Series 413
14 4 5 Solvability and the Derived Series 413
14 4 6 Adjunction of Roots of Unity by Radicals 414
Exercise Set 14 4 415
14 5 Proof of Necessity of Solvability for Solutions in Radicals 415
14 5 1 Necessary Condition for Solutions in Radicals 415
14 5 2 Tower Adjunction 416
XX CONTENTS
14 5 3 Generators for the Symmetric Group 417
14 5 4 Some Polynomials with Symmetric Groups 418
14 55A Polynomial Not Solvable in Radicals 419
Exercise Set 14 5 420
Chapter 15 LINEAR TRANSFORMATIONS
ON VECTOR SPACES 421
15 1 Linear Transformations 421
15 1 1 Linear Transformations 421
15 1 2 The Matrix of a Linear Transformation 422
15 1 3 Linear Transformations and Matrices in V2 and V3 423
15 1 4 The Correspondence Between Linear Transformations
and Matrices 424
15 1 5 Matrix-Vector Multiplication 426
15 1 6 Range Space and Column Space 426
15 1 7 Addition of Linear Transformations and Matrices 427
15 1 8 Composition of Linear Transformations and Multiplication
of Matrices 428
15 1 9 The Ring of Linear Transformations 430
15 1 10 Change of Basis and Similar Matrices 433
15 1 11 A Change of Basis in V3(Q) 434
Exercise Set 15 1 436
15 2 Determinants 438
15 2 1 Determinants over a Commutative Ring 438
15 2 2 Elementary Properties of Determinants 438
15 2 3 Product of Matrices and the Product of Determinants 439
Exercise Set 15 2 442
Chapter 16 MODULES AND CANONICAL FORMS 443
16 1 Introduction to Modules 443
16 1 1 Module over a Ring 444
16 1 2 Examples of Modules 444
16 1 3 An Abelian Group Is a Z-module 445
16 14A Vector Space over F Is an F[x]-module 445
16 1 5 Cyclic Module 446
16 1 6 Submodule 446
16 1 7 Submoduks of Cyclic Modules 447
16 1 8 Homomorphisms and Direct Sums of Modules 447
CONTENTS XXi
16 1 9 Module Decompositions of the Cyclic Group of Order 6 as a
Z-module 448
16 1 10 The Vector Space V2(Q) as a Cyclic Module over
Q[x] 449
16 1 11 Order of a Module 450
16 1 12 The Order of the Cyclic Q[x]-Module of
Example 16 1 10 450
16 1 13 Submodules of a Cyclic Module 451
Exercise Set 16 1 452
16 2 Finitely Generated Modules 453
16 2 1 Finitely Generated Modules 453
16 2 2 Finite Generation Is Inherited 454
16 2 3 Free Modules 455
16 2 4 Representation of Free Modules 455
16 2 5 Homomorphism Property of Free Modules 455
16 26A Property of Matrices over a Commutative Ring 456
16 2 7 Exchange Property for Free Modules 456
16 2 8 The Main Algorithm for Choosing Generators for a
Submodule 458
16 29A Submodule of Z © Z 463
16 2 10 Diagonal Form for Matrices 466
Exercise Set 16 2 466
16 3 Module Decomposition 467
16 3 1 The Existence of the Canonical Direct Sum
Decomposition 467
16 3 2 Decomposition of Abelian Groups 469
16 3 3 Finitely Presented Modules 469
16 3 4 Abelian Group Example 470
16 3 5 The Decomposition of the Cyclic Q[t]-module
of Example 16 1 10 472
Exercise Set 16 3 474
16 4 Uniqueness of Module Decomposition 474
16 4 1 Uniqueness of the Direct Sum Decomposition 475
16 4 2 Cancellation of a Cyclic Summand 476
16 4 3 Proof of the Uniqueness Theorem 16 4 1 480
16 4 4 Uniqueness of Direct Sum Decomposition
for Abelian Groups 481
16 4 5 Determination of the Abelian Groups of Order n 481
16 4 6 The Abelian Groups of Order 2250 482
Exercise Set 16 4 484
xxii CONTENTS
16 5 Canonical Forms for a Linear Transformation 484
16 5 1 Cyclic Submodule as a Subspace 486
16 5 2 Rational Canonical Form and Invariant Factors 488
16 5 3 Determination of the Invariant Factors of a Linear
Transformation 489
16 5 4 Characteristic Polynomial, Invariant Factors, and the
Hamilton-Cay ley Theorem 491
16 5 5 Test for Similarity of Matrices 492
16 5 6 Determination of the Invariant Factors of a Matrix
over Q 492
16 5 7 Decomposition into Cyclic Subspaces Whose Orders Are
Prime Powers 494
16 5 8 Representation of a Matrix as a Direct Sum of Companion
Matrices 495
16 5 9 Existence of the Jordan Canonical Form 496
16 5 10 Uniqueness of the Jordan Canonical Form 497
Exercise Set 16 5 498
Answers to Selected Problems
Index
|
| any_adam_object | 1 |
| author | Dean, Richard A. |
| author_facet | Dean, Richard A. |
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| building | Verbundindex |
| bvnumber | BV005736483 |
| callnumber-first | Q - Science |
| callnumber-label | QA162 |
| callnumber-raw | QA162 |
| callnumber-search | QA162 |
| callnumber-sort | QA 3162 |
| callnumber-subject | QA - Mathematics |
| classification_tum | MAT 110f |
| ctrlnum | (OCoLC)20690379 (DE-599)BVBBV005736483 |
| dewey-full | 512/.02 |
| dewey-hundreds | 500 - Natural sciences and mathematics |
| dewey-ones | 512 - Algebra |
| dewey-raw | 512/.02 |
| dewey-search | 512/.02 |
| dewey-sort | 3512 12 |
| dewey-tens | 510 - Mathematics |
| discipline | Mathematik |
| format | Book |
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| id | DE-604.BV005736483 |
| illustrated | Illustrated |
| indexdate | 2024-07-09T16:33:49Z |
| institution | BVB |
| isbn | 0060416017 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-003580632 |
| oclc_num | 20690379 |
| open_access_boolean | |
| owner | DE-91 DE-BY-TUM DE-188 |
| owner_facet | DE-91 DE-BY-TUM DE-188 |
| physical | XXIX, 524 S. graph. Darst. |
| publishDate | 1990 |
| publishDateSearch | 1990 |
| publishDateSort | 1990 |
| publisher | Harper & Row |
| record_format | marc |
| spelling | Dean, Richard A. Verfasser aut Classical abstract algebra Richard A. Dean New York u.a. Harper & Row 1990 XXIX, 524 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier Algebra, Abstract Algebra (DE-588)4001156-2 gnd rswk-swf Universelle Algebra (DE-588)4061777-4 gnd rswk-swf Algebra (DE-588)4001156-2 s DE-604 Universelle Algebra (DE-588)4061777-4 s DE-188 HEBIS Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=003580632&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Dean, Richard A. Classical abstract algebra Algebra, Abstract Algebra (DE-588)4001156-2 gnd Universelle Algebra (DE-588)4061777-4 gnd |
| subject_GND | (DE-588)4001156-2 (DE-588)4061777-4 |
| title | Classical abstract algebra |
| title_auth | Classical abstract algebra |
| title_exact_search | Classical abstract algebra |
| title_full | Classical abstract algebra Richard A. Dean |
| title_fullStr | Classical abstract algebra Richard A. Dean |
| title_full_unstemmed | Classical abstract algebra Richard A. Dean |
| title_short | Classical abstract algebra |
| title_sort | classical abstract algebra |
| topic | Algebra, Abstract Algebra (DE-588)4001156-2 gnd Universelle Algebra (DE-588)4061777-4 gnd |
| topic_facet | Algebra, Abstract Algebra Universelle Algebra |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=003580632&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
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