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E-grāmata: A Survey of Modern Algebra

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  • Formāts: 512 pages
  • Izdošanas datums: 19-Dec-2017
  • Izdevniecība: A K Peters
  • ISBN-13: 9781439864531
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A Survey of Modern AlgebraPalielināt
  • Formāts: 512 pages
  • Izdošanas datums: 19-Dec-2017
  • Izdevniecība: A K Peters
  • ISBN-13: 9781439864531
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Now available in paperback, this classic, written by two young instructors who became giants in their field, has shaped the understanding of modern algebra for generations of mathematicians and remains a valuable reference and text for self study and college courses.

This classic, written by two young instructors who became giants in their field, has shaped the understanding of modern algebra for generations of mathematicians and remains a valuable reference and text for self study and college courses.
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 Divisibility
16
1.7 The Euclidean Algorithm
18
1.8 Fundamental Theorem of Arithmetic
23
1.9 Congruences
25
1.10 The Rings Z„
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 Spaces 168
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 Fields 420
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
Index 489
Birkhoff , Garrett; Mac Lane , Saunders
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