书籍详情
近世代数概论（英文版·第5版）
作者：（美）伯克霍夫、等
出版社：人民邮电出版社
出版时间：2007-08-01
ISBN：9787115162311
定价：¥69.00
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内容简介
本书出自近世代数领域的两位科学巨匠之手，是一本经典的教材。全书共分为15章，内容包括：整数、多项式、实数、复数、矩阵代数、线性群、行列式和标准型、布尔代数和格、超限算术、环和理想、代数数域和伽罗华理论等。 本书曾帮助过几代人理解近世代数，至今仍是一本非常有价值的参考书和教材，适合数学专业及其他理工科专业高年级本科生和研究生使用。 本书出自近世代数领域的两位科学巨匠之手，是一本经典的教材。全书共分为15章，内容包括：整数、多项式、实数、复数、矩阵代数、线性群、行列式和标准型、布尔代数和格、超限算术、环和理想、代数数域和伽罗华理论等。 本书曾帮助过几代人理解近世代数，至今仍是一本非常有价值的参考书和教材，适合数学专业及其他理工科专业高年级本科生和研究生使用。
作者简介
Garett Birkhoff（1911-1996）已故世界著名数学家，生前曾任国际数学家大会组织委员会主席、美国数学会副主席，美国工业与应用数学会主席、《大不列颠百科全书》编委，美国科学院院士，哈佛大学教授，1933年开创格论研究，使其成为数学的一个重要分文。
目录
Preface to the Fourth Edition
1 The Integers 1
1.1 Commutative Rings; Integral Domains 1
1.2 Elementary Properties of Commutative Rings 3
1.3 Ordered Domains 8
1.4 Well-Ordering Principle 11
1.5 Finite Induction; Laws of Exponents 12
1.6 ivisibility 16
1.7 The Euclidean Algorithm 18
1.8 Fundamental Theorem of Arithmetic 23
1.9 Congruences 25
1.10 The Rings Zn 29
1.11 Sets, Functions, and Relations 32
1.12 Isomorphisms and Automorphisms 35
2 Rational Numbers and Fields 38
2.1 Definition of a Field 38
2.2 Construction of the Rationals 42
2.3 Simultaneous Linear Equations 47
2.4 Ordered Fields 52
2.5 Postulates for the Positive Integers 54
2.6 Peano Postulates 57
3 Polynomials 61
3.1 Polynomial Forms 61
3.2 Polynomial Functions 65
3.3 Homomorphisms of Commutative Rings 69
3.4 Polynomials in Several Variables 72
3.5 The Division Algorithm 74
3.6 Units and Associates 76
3.7 Irreducible Polynomials 78
3.8 Unique Factorization Theorem 80
3.9 Other Domains with Unique Factorization 84
3.10 Eisenstein's Irreducibility Criterion 88
3.11 Partial Fractions 90
4 Real Numbers 94
4.1 Dilemma of Pythagoras 94
4.2 Upper and Lower Bounds 96
4.3 Postulates for Real Numbers 98
4.4 Roots of Polynomial Equations 101
4.5 Dedekind Cuts 104
5 Complex Numbers 107
5.1 Definition 107
5.2 The Complex Plane 110
5.3 Fundamental Theorem of Algebra 113
5.4 Conjugate Numbers and Real Polynomials 117
5.5 Quadratic and Cubic Equations 118
5.6 Solution of Quartic by Radicals 121
5.7 Equations of Stable Type 122
6 Groups 124
6.1 Symmetries of the Square 124
6.2 Groups of Transformations 126
6.3 Further Examples 131
6.4 Abstract Groups 133
6.5 Isomorphism 137
6.6 Cyclic Groups 140
6.7 Subgroups 143
6.8 Lagrange's Theorem 146
6.9 Permutation Groups 150
6.10 Even and Odd Permutations 153
6.11 Homomorphisms 155
6.12 Automorphisms; Conjugate Elements 157
6.13 Quotient Groups 161
6.14 Equivalence and Congruence Relations 164
7 Vectors and Vector Spaces168
7.1 Vectors in a Plane 168
7.2 Generalizations 169
7.3 Vector Spaces and Subspaces 171
7.4 Linear Independence and Dimension 176
7.5 Matrices and Row-equivalence 180
7.6 Tests for Linear Dependence 183
7.7 Vector Equations; Homogeneous Equations 188
7.8 Bases and Coordinate Systems 193
7.9 Inner Products 198
7.10 Euclidean Vector Spaces 200
7.11 Normal Orthogonal Bases 203
7.12 Quotient-spaces 206
7.13 Linear Functions and Dual Spaces 208
8 The Algebra of Matrices 214
8.1 Linear Transformations and Matrices 214
8.2 Matrix Addition 220
8.3 Matrix Multiplication 222
8.4 Diagonal, Permutation, and Triangular Matrices 228
8.5 Rectangular Matrices 230
8.6 Inverses 235
8.7 Rank and Nullity 241
8.8 Elementary Matrices 243
8.9 Equivalence and Canonical Form 248
8.10 Bilinear Functions and Tensor Products 251
8.11 Quaternions 255
9 Linear Groups 260
9.1 Change of Basis 260
9.2 Similar Matrices and Eigenvectors 263
9.3 The Full Linear and Affine Groups 268
9.4 The Orthogonal and Euclidean Groups 272
9.5 Invariants and Canonical Forms 277
9.6 Linear and Bilinear Forms 280
9.7 Quadratic Forms 283
9.8 Quadratic Forms Under the Full Linear Group 286
9.9 Real Quadratic Forms Under the Full Linear Group 288
9.10 Quadratic Forms Under the Orthogonal Group 292
9.11 Quadrics Under the Affine and Euclidean Groups 296
9.12 Unitary and Hermitian Matrices 300
9.13 Affine Geometry 305
9.14 Projective Geometry 312
10 Determinants and Canonical Forms 318
10.1 Definition and Elementary Properties of Determinants 318
10.2 Products of Determinants 323
10.3 Determinants as Volumes 327
10.4 The Characteristic Polynomial 331
10.5 The Minimal Polynomial 336
10.6 Cayley-Hamilton Theorem 340
10.7 Invariant Subspaces and Reducibility 342
10.8 First Decomposition Theorem 346
10.9 Second Decomposition Theorem 349
10.10 Rational and Jordan Canonical Forms 352
11 Boolean Algebras and Lattices 357
11.1 Basic Definition 357
11.2 Laws: Analogy with Arithmetic 359
11.3 Boolean Algebra 361
11.4 Deduction of Other Basic Laws 364
11.5 Canonical Forms of Boolean Polynomials 368
11.6 Partial Orderings 371
11.7 Lattices 374
11.8 Representation by Sets 377
12 Transfinite Arithmetic 381
12.1 Numbers and Sets 381
12.2 Countable Sets 383
12.3 Other Cardinal Numbers 386
12.4 Addition and Multiplication of Cardinals 390
12.5 Exponentiation 392
13 Rings and Ideals 395
13.1 Rings 395
13.2 Homomorphisms 399
13.3 Quotient-rings 403
13.4 Algebra of Ideals 407
13.5 Polynomial Ideals 410
13.6 Ideals in Linear Algebras 413
13.7 The Characteristic of a Ring 415
13.8 Characteristics of Fields 418
14 Algebraic Number Fields420
14.1 Algebraic and Transcendental Extensions 420
14.2 Elements Algebraic over a Field 423
14.3 Adjunction of Roots 425
14.4 Degrees and Finite Extensions 429
14.5 Iterated Algebraic Extensions 431
14.6 Algebraic Numbers 435
14.7 Gaussian Integers 439
14.8 Algebraic Integers 443
14.9 Sums and Products of Integers 445
14.10 Factorization of Quadratic Integers 448
15 Galois Theory 452
15.1 Root Fields for Equations 452
15.2 Uniqueness Theorem 454
15.3 Finite Fields 456
15.4 The Galois Group 459
15.5 Separable and Inseparable Polynomials 464
15.6 Properties of the Galois Group 467
15.7 Subgroups and Subfields 471
15.8 Irreducible Cubic Equations 474
15.9 Insolvability of Quintic Equations 478
Bibliography 483
List of Special Symbols 486
Index489
1 The Integers 1
1.1 Commutative Rings; Integral Domains 1
1.2 Elementary Properties of Commutative Rings 3
1.3 Ordered Domains 8
1.4 Well-Ordering Principle 11
1.5 Finite Induction; Laws of Exponents 12
1.6 ivisibility 16
1.7 The Euclidean Algorithm 18
1.8 Fundamental Theorem of Arithmetic 23
1.9 Congruences 25
1.10 The Rings Zn 29
1.11 Sets, Functions, and Relations 32
1.12 Isomorphisms and Automorphisms 35
2 Rational Numbers and Fields 38
2.1 Definition of a Field 38
2.2 Construction of the Rationals 42
2.3 Simultaneous Linear Equations 47
2.4 Ordered Fields 52
2.5 Postulates for the Positive Integers 54
2.6 Peano Postulates 57
3 Polynomials 61
3.1 Polynomial Forms 61
3.2 Polynomial Functions 65
3.3 Homomorphisms of Commutative Rings 69
3.4 Polynomials in Several Variables 72
3.5 The Division Algorithm 74
3.6 Units and Associates 76
3.7 Irreducible Polynomials 78
3.8 Unique Factorization Theorem 80
3.9 Other Domains with Unique Factorization 84
3.10 Eisenstein's Irreducibility Criterion 88
3.11 Partial Fractions 90
4 Real Numbers 94
4.1 Dilemma of Pythagoras 94
4.2 Upper and Lower Bounds 96
4.3 Postulates for Real Numbers 98
4.4 Roots of Polynomial Equations 101
4.5 Dedekind Cuts 104
5 Complex Numbers 107
5.1 Definition 107
5.2 The Complex Plane 110
5.3 Fundamental Theorem of Algebra 113
5.4 Conjugate Numbers and Real Polynomials 117
5.5 Quadratic and Cubic Equations 118
5.6 Solution of Quartic by Radicals 121
5.7 Equations of Stable Type 122
6 Groups 124
6.1 Symmetries of the Square 124
6.2 Groups of Transformations 126
6.3 Further Examples 131
6.4 Abstract Groups 133
6.5 Isomorphism 137
6.6 Cyclic Groups 140
6.7 Subgroups 143
6.8 Lagrange's Theorem 146
6.9 Permutation Groups 150
6.10 Even and Odd Permutations 153
6.11 Homomorphisms 155
6.12 Automorphisms; Conjugate Elements 157
6.13 Quotient Groups 161
6.14 Equivalence and Congruence Relations 164
7 Vectors and Vector Spaces168
7.1 Vectors in a Plane 168
7.2 Generalizations 169
7.3 Vector Spaces and Subspaces 171
7.4 Linear Independence and Dimension 176
7.5 Matrices and Row-equivalence 180
7.6 Tests for Linear Dependence 183
7.7 Vector Equations; Homogeneous Equations 188
7.8 Bases and Coordinate Systems 193
7.9 Inner Products 198
7.10 Euclidean Vector Spaces 200
7.11 Normal Orthogonal Bases 203
7.12 Quotient-spaces 206
7.13 Linear Functions and Dual Spaces 208
8 The Algebra of Matrices 214
8.1 Linear Transformations and Matrices 214
8.2 Matrix Addition 220
8.3 Matrix Multiplication 222
8.4 Diagonal, Permutation, and Triangular Matrices 228
8.5 Rectangular Matrices 230
8.6 Inverses 235
8.7 Rank and Nullity 241
8.8 Elementary Matrices 243
8.9 Equivalence and Canonical Form 248
8.10 Bilinear Functions and Tensor Products 251
8.11 Quaternions 255
9 Linear Groups 260
9.1 Change of Basis 260
9.2 Similar Matrices and Eigenvectors 263
9.3 The Full Linear and Affine Groups 268
9.4 The Orthogonal and Euclidean Groups 272
9.5 Invariants and Canonical Forms 277
9.6 Linear and Bilinear Forms 280
9.7 Quadratic Forms 283
9.8 Quadratic Forms Under the Full Linear Group 286
9.9 Real Quadratic Forms Under the Full Linear Group 288
9.10 Quadratic Forms Under the Orthogonal Group 292
9.11 Quadrics Under the Affine and Euclidean Groups 296
9.12 Unitary and Hermitian Matrices 300
9.13 Affine Geometry 305
9.14 Projective Geometry 312
10 Determinants and Canonical Forms 318
10.1 Definition and Elementary Properties of Determinants 318
10.2 Products of Determinants 323
10.3 Determinants as Volumes 327
10.4 The Characteristic Polynomial 331
10.5 The Minimal Polynomial 336
10.6 Cayley-Hamilton Theorem 340
10.7 Invariant Subspaces and Reducibility 342
10.8 First Decomposition Theorem 346
10.9 Second Decomposition Theorem 349
10.10 Rational and Jordan Canonical Forms 352
11 Boolean Algebras and Lattices 357
11.1 Basic Definition 357
11.2 Laws: Analogy with Arithmetic 359
11.3 Boolean Algebra 361
11.4 Deduction of Other Basic Laws 364
11.5 Canonical Forms of Boolean Polynomials 368
11.6 Partial Orderings 371
11.7 Lattices 374
11.8 Representation by Sets 377
12 Transfinite Arithmetic 381
12.1 Numbers and Sets 381
12.2 Countable Sets 383
12.3 Other Cardinal Numbers 386
12.4 Addition and Multiplication of Cardinals 390
12.5 Exponentiation 392
13 Rings and Ideals 395
13.1 Rings 395
13.2 Homomorphisms 399
13.3 Quotient-rings 403
13.4 Algebra of Ideals 407
13.5 Polynomial Ideals 410
13.6 Ideals in Linear Algebras 413
13.7 The Characteristic of a Ring 415
13.8 Characteristics of Fields 418
14 Algebraic Number Fields420
14.1 Algebraic and Transcendental Extensions 420
14.2 Elements Algebraic over a Field 423
14.3 Adjunction of Roots 425
14.4 Degrees and Finite Extensions 429
14.5 Iterated Algebraic Extensions 431
14.6 Algebraic Numbers 435
14.7 Gaussian Integers 439
14.8 Algebraic Integers 443
14.9 Sums and Products of Integers 445
14.10 Factorization of Quadratic Integers 448
15 Galois Theory 452
15.1 Root Fields for Equations 452
15.2 Uniqueness Theorem 454
15.3 Finite Fields 456
15.4 The Galois Group 459
15.5 Separable and Inseparable Polynomials 464
15.6 Properties of the Galois Group 467
15.7 Subgroups and Subfields 471
15.8 Irreducible Cubic Equations 474
15.9 Insolvability of Quintic Equations 478
Bibliography 483
List of Special Symbols 486
Index489
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