CONTENTS
Volume Ⅰ
Preface = ⅴ
VECTOR SPACES
1. First introduction : Affine geometry = 1
Examples from geometry = 1
Vectors as equivalence classes of arrows = 2
Addition and scalar multiplication = 3
Expressing geometrical concepts by vectors = 7
Applications of the vector concept to geometrical problems = 9
Coordinate systems = 15
Reformulating geometrical problems as systems of linear equations = 17
Exercises = 19
2. Second introduction : Linear equations = 22
Examples from electrical engineering and economics = 22
Fields = 24
Matrices = 25
Matrix operations = 26
The Gaussian algorithm = 30
Matrix calculus = 33
Matrices as linear mappings = 37
Homogeneous and inhomogeneous systems of linear equations = 40
Exercises = 42
3. Vector spaces = 49
Vector spaces and subspaces = 49
Examples = 50
Free and generating sets = 54
Bases = 55
Dimension = 58
Dimension formula for subspaces = 58
Direct decompositions = 59
Quotient spaces = 62
Dimension formula for quotient spaces = 63
Exercises = 65
4. Linear and affine mappings = 72
Characterization of linear and affine mappings = 72
Examples = 74
Invariant subspaces = 76
Semisimplicity = 77
Isomorphisms = 80
Image, kernel and cokernel = 81
Homomorphism theorem = 82
Dimension formula for linear mappings = 83
Vector spaces of linear mappings = 85
Dual spaces and dual mappings = 86
Exercises = 88
5. Abstract affine geometry = 95
Affine spaces and subspaces = 95
Dimension formula for affine subspaces = 99
Affine mappings = 100
Convexity = 101
The natural topology of a finite-dimensional real or complex vector space Norms = 103
Norms = 104
Exercises = 105
6. Representation of linear mappings by matrices = 111
Parameterizations, basis isomorphisms and coordinate transformations = 111
Matrix representations of linear mappings = 112
Matrix inversion = 114
Conceptual meaning of the transpose = 117
Conceptual meaning of block decompositions = 118
Change of basis = 119
Equivalence of matrices = 121
Rank of a matrix = 121
Classification of matrices up to equivalence = 122
Similarity of square matrices = 124
Canonical forms for projections, reflections and nilpotent endomorphisms = 124
Exercises = 130
7. Determinants = 138
Determinants as natural constructions in elimination theory = 138
Existence and uniqueness of determinant functions = 142
Significance of determinants for systems of linear equations = 145
Properties of determinants = 146
Examples = 147
The adjunct of a matrix = 150
Cramer's rule = 152
Exercises = 154
8. Volume functions = 159
Motivation : Oriented volumes = 159
Volume functions = 160
Existence and uniqueness = 162
Closed form for the determinant of a matrix = 164
Expressing minors of a matrix product by minors of the individual factors = 165
Determinant of an endomorphism as a distortion factor for volumes = 166
Trace of an endomorphism = 168
Basis deformations = 170
Orientations = 170
Exercises = 173
9. Eigenvectors and eigenvalues = 177
Definition of eigenvectors and eigenvalues = 177
Linear independence of eigenvectors = 178
Eigenspaces and generalized eigenspaces = 178
Characteristic polynomial = 179
Gershgorin's theorem = 183
Geometric and algebraic multiplicity = 185
Diagonalizabilily = 186
Simultaneous diagonalizability = 187
Triagonalizability = 188
Simultaneous triagonalizability = 189
Spectral mapping theorem = 190
Perron-Frobenius theorem = 194
Exercises = 197
10. Classification of endomorphisms up to similarity = 205
Algebras = 205
Polynomial expressions of an endomorphism = 205
Matrix polynomials = 208
Hamilton-Cayley theorem = 209
Minimal polynomial = 210
Jordan canonical form = 211
Additive and multiplicative Jordan decomposition = 215
Frobenius' theorem = 217
Determinantal and elementary divisors = 219
Exercises = 223
11. Tensor products and base - field extensions = 230
Complexification = 230
Tensor product of vector spaces = 231
Properties = 232
Base-field extensions = 238
Fitting decomposition = 240
Seimsimplicity = 241
Jordan decomposition over fields which are not algebraically closed = 243
Exercises = 245
12. Metric geometry = 249
Cartesian coordinate systems = 249
Scalar product = 249
Applications in geometry = 252
Oriented areas and volumes = 255
Vector product = 258
Applications in geometry and mechanics = 260
Exercises = 268
13. Euclidean spaces = 273
Scalar products = 273
Orthogonality = 276
Orthogonal decompositions = 279
Orthonormalization = 281
Riesz' representation theorem = 282
Gram's determinant = 283
Normed volume functions = 284
Measure theory = 285
Exercises = 296
14. Linear mappings between Euclidean spaces = 303
Adjoint of a linear mapping = 303
Properties = 304
Trace and operator norm = 306
Normal, self-adjoint, skew-adjoint, orthogonal and unitary endomorphisms = 308
Characterization = 308
Canonical forms = 312
Positive definite endomorphisms = 315
Pseudoinverses = 318
Kalman filter = 322
Exercises = 324
15. Bilinear forms = 331
Bilinear and sesquilinear forms = 331
Representation by matrices = 332
Congruence of matrices = 333
Symmetric, antisymmetric and alternating forms = 335
Orthogonality = 336
Isotropy, rank and degeneracy = 337
Orthogonal complements = 337
Classification of alternating forms = 341
Pfaffian = 342
Classification of real symmetric forms by rank and signature = 344
Quadratic forms and quadrics = 347
Metrical classification of real symmetric forms ; Sylvester's law of inertia = 349
Hyperbolic planes = 352
Witt's theorem = 354
Bruck-Chowla-Ryser theorem = 356
Exercises = 358
16. Groups of automorphisms = 366
Linear transformation groups = 366
Examples = 366
Structure of orthogonal groups = 370
Group-theoretical interpretation of the Gaussian algorithm = 373
Gaussian decomposition of GL (n, K) = 374
Bruhat decomposition of GL (n, K) = 376
Iwasawa, polar and Cartan decomposition of GL (n, R) and GL (n, C) = 378
Parameterizations of automorphism groups = 380
Cayley transform = 382
Exercises = 384
17. Application : Markov chains = 393
Markov processes = 393
Transition matrices and graphs = 393
Examples = 394
Long-term probabilities = 396
Communicating classes = 399
Canonical form of a Markov process = 400
Fundamental matrix = 401
Regular Markov chains = 403
Ergodic Markov chains = 405
Cyclic classes = 407
Mean first passage matrix = 411
Examples in genetics = 413
Exercises = 419
18. Application : Matrix calculus and differential equations = 424
Examples from electrical Engineering and economics = 424
Existence and uniqueness of solutions for linear initial value problems = 426
Wronskian determinant = 427
Fundamental system = 428
Exponential function for matrices = 430
Matrix differential equations with constant coefficients = 432
Higher-order differential equations = 435
Characteristic polynomial = 438
Fundamental system = 438
Method of judicious guessing = 440
Fulmer's algorithm to exponentiate a matrix = 442
Exercises = 444
GROUPS
19. Introduction : Symmetries of geometric figures = 451
Symmetries in art and nature = 451
Symmetry transformations = 453
Types of one- and two-dimensional ornaments = 454
Exercises = 460
20. Groups = 461
Semigroups, monoids and groups = 461
Elementary properties = 462
Group table = 463
Integers = 463
Additive and multiplicative group of a field = 463
Additive group of a vector space = 463
Permutation groups = 463
Symmetric groups = 464
Dihedral groups = 464
Automorphism group of a graph = 465
Matrix groups = 466
Groups of residue classes = 468
Direct products = 470
Exercises = 472
21. Subgroups and cosets = 476
Subgroups = 476
Centralizers and normalizers = 478
Subgroup generated by a set = 478
Cyclic groups = 480
Order of an element = 480
Cosets = 486
Index of a subgroup = 486
Lagrange's theorem = 487
Applications in number theory = 488
Exercises = 489
22. Symmetric and alternating groups = 497
Cycle decomposition = 498
Cycle-structure and conjugacy = 498
Generating sets for Symn = 500
Even and odd permutations = 503
Alternating groups = 503
Exercises = 505
23. Group homomorphisms = 511
Isomorphisms = 512
Conjugation ; inner and outer automorphisms = 514
Homomorphisms = 515
Kernel and image = 518
Embeddings = 521
Cayley's theorem = 521
Representations = 522
Exercises = 523
24. Normal subgroups and factor groups = 529
Characteristic and normal subgroups = 529
Quotient groups = 531
Isomorphism theorems = 532
Abelianization of a group = 535
Simple groups = 536
Simplicity of the alternating groups = 536
Semidirect products = 537
Exercises = 541
25. Free groups ; generators and relations = 548
Free groups = 549
Universal property = 550
Generators and relations = 553
Von Dyck's theorem = 553
Free abelian groups = 554
Universal property = 555
Linearly independent and generating sets ; bases = 555
Direct sums = 556
Rank of a free abelian group = 557
Torsion subgroup of an abelian group = 557
Classification of finitely generated abelian groups = 561
Character group of an abelian group = 567
Exercises = 570
26. Group actions = 575
Group actions = 575
Examples = 575
Orbits and stabilizers = 577
Transitivity = 577
Automorphism groups of graphs = 580
Burnside's lemma = 581
P o' lya's theorem = 587
Applications in combinatorics = 588
Exercises = 591
27. Group - theoretical applications of group actions = 596
Actions of groups and coset spaces = 596
Class equation = 597
Deriving structural properties of a finite group from its order = 597
Sylow's theorems = 600
Applications = 602
Exercises = 604
28. Nilpotent and solvable groups = 608
Commutators = 608
Commutator series and descending central series = 608
Examples = 609
Solvability and nilpotency = 613
Ascending central series = 615
Characterization of finite nilpotent groups = 617
Affine subgroups of Symn = 617
Characterization of the solvable transitive subgroups of Symp = 620
Exercises = 622
29. Topological methods in group theory = 624
Topological groups and their subgroups = 624
Continuous homomorphisms = 628
Isomorphism theorems = 630
Baire's theorem = 633
Open mapping theorem = 633
Identity component = 635
Examples = 635
Krull topology = 639
Exercises = 641
30. Analytical methods in group theory = 644
Tangent space of a matrix group = 645
One-parameter subgroups = 647
Lie object of a matrix group = 648
Lie algebras = 650
Examples = 650
Relation between a closed matrix group and its Lie algebra = 654
Adjoint action = 655
Baker-Campbell-Hausdorff formula = 660
Analytic matrix groups and their Lie group topology = 662
Exercises = 666
31. Groups in topology = 671
Basic ideas in algebraic topology = 671
Homotopy = 673
Homotopy groups = 677
Fixed point theorems = 679
Commutativity of the higher homotopy groups = 682
Homology = 683
Homology groups = 688
H1 (X) as the abelianization of π1 (X) = 689
Knot groups = 693
Wirtinger presentation = 693
Exercises = 697
Appendix = 701
A. Sets and functions = 701
B. Relations = 709
C. Cardinal and ordinal number = 718
D. Point-set topology = 728
E. Categories and functors = 737
Bibliography = 743
Index = 745
Volume Ⅱ
Preface = v
RINGS AND FIELDS
1. Introduction : The art of doing arithmetic = 1
Euler's and Fermat's theorem = 2
Divisibility rules = 2
Other examples for the use of congruence classes to obtain number-theoretical results = 3
Wilson's theorem = 4
Exercises = 6
2. Rings and ring homomorphisms = 10
Rings, commutative rings and unital rings = 10
Subrings = 11
Examples = 11
Power series rings = 12
Polynomial rings = 13
Matrix rings = 15
Rings of functions = 15
Convolution rings = 16
Direct products and sums = 17
Ring homomorphisms, isomorphisms and embeddings = 17
Exercises = 20
3. Integral domains and fields = 29
Zero-divisors = 29
Nilpotent elements = 29
Units = 31
Examples = 31
Divisibility = 35
Integral domains = 36
Fields and skew-fields = 38
Quotient fields = 40
Application : Mikusinski's operator calculus = 41
Exercises = 48
4. Polynomial and power series rings = 54
Polynomials in one or more variables = 54
Division with remainder = 57
Roots and their multiplicities = 58
Derivatives = 60
Symmetric polynomials = 61
Main theorem on symmetric polynomials = 64
Discriminant = 65
Exercises = 67
5. Ideals and quotient rings = 76
Ideals = 76
Ideals generated by a set = 77
Principal ideals = 77
Simple rings = 78
Quotient rings = 79
Isomorphism theorems = 80
Maximal ideals = 82
Chinese remainder theorem = 83
Exercises = 86
6. Ideals in commutative rings = 94
Prime and primary ideals = 94
Ideal quotients = 96
Radical of an ideal ; radical ideal = 96
Primary decompositions = 102
Symbolic powers = 105
Exercises = 107
7. Factorization in integral domains = 112
Prime and irreducible elements = 112
Expressing notions of divisibility in terms of ideals = 114
Factorization domains and unique factorization domains = 116
Characterization of principal ideal domains = 119
Euclidean domains = 119
Examples = 120
Euclidean algorithm = 125
Exercises = 128
8. Factorization in polynomial and power series rings = 135
Transmission of the unique factorization property from R to R( x1 ,…, xn ) = 135
Unique factorization properly for power series rings = 139
Factorization algorithms for polynomials = 142
Irreducibility criteria for polynomials = 147
Resultant of two polynomials = 150
Decomposition of homogeneous polynomials = 154
Exercises = 155
9. Number-theoretical applications of unique factorization = 163
Representability of prime numbers by quadratic forms = 163
Legendre symbol = 164
Quadratic reciprocity law = 166
Three theorems of Fermat = 169
Ramanujan-Nagell theorem = 172
Insolvability of the equation x3 + y3 = z3 inN = 174
Kummer's theorem = 178
Exercises = 180
10. Modules and integral ring extensions = 185
Modules = 185
Free modules = 185
Submodules and quotient modules = 186
Module homomorphisms = 187
Noetherian and Artinian modules = 187
Algebras over commutative rings = 191
Algebraic and integral ring extensions = 191
Module-theoretical characterization of integral elements = 193
Integral closure = 194
Exercises = 196
11. Noetherian rings = 203
Characterization of Noetherian rings = 203
Examples = 203
Hilbert's Basis Theorem = 205
Cohen's theorem = 205
Noetherian induction = 208
Primary decompositions in Noetherian rings = 209
Artinian rings = 212
Exercises = 215
12. Field extensions = 217
Field extensions = 217
Intermediate fields = 217
Minimal polynomial = 219
Degree of simple extensions = 220
Degree formula = 221
L u ·· roth's theorem = 223
Cardinality of finite fields = 224
Trace and norm of finite field extensions = 225
Algebraic extensions = 226
Characterization of simple extensions = 227
Algebraically generating and independent sets = 228
Transcendence bases = 229
Transcendence degree = 231
Decomposing a field extension into an algebraic and a purely transcendental extension = 232
Zariski's lemma = 232
Ruler and compass constructions = 233
Exercises = 238
13. Splitting fields and normal extensions = 248
Adjoining roots of polynomials = 248
Splitting fields = 250
Normal extensions = 250
Extensions of field isomorphisms = 250
Algebraic closures = 253
Conjugates of an algebraic element = 256
Normal closures = 257
Finite fields = 259
Exercises = 262
14. Separability of field extensions = 268
Separability = 268
Perfect fields = 269
Existence of primitive elements = 271
Characterization of separability via embeddings = 272
Properties of separable extensions = 273
Pure inseparability = 275
Decomposition of an algebraic field extension into a separable and a purely inseparable extension = 277
Degree of separability and inseparability = 278
Trace and norm = 279
Discriminant of a finite field extension = 283
Characterization of separability via the trace = 285
Exercises = 287
15. Field theory and integral ring extensions = 291
Trace, norm and minimal polynomial of an integral element = 292
Integral closure of Z in quadratic number fields = 293
Existence of integral bases = 295
Discriminant of an algebraic number field = 296
Stickelberger's theorem = 296
Integral closure of Z in special cyclotomic fields = 297
Exercises = 299
16. Affine algebras = 301
Affine algebras = 301
Noether's normalization theorem = 302
Zariski's lemma = 304
Stronger versions of Noether's normalization theorem = 306
Krull dimension of a commutative ring = 308
Lying-over theorem = 309
Going-up theorem = 310
Going-down theorem = 312
Krull dimension of an affine algebra = 313
Exercises = 315
17. Ring theory and algebraic geometry = 318
Affine varieties = 318
Correspondence between varieties and ideals = 322
Hilbert's Null-stellensatz = 324
Zariski topology = 326
Irreducible varieties = 327
Polynomial and rational mappings between varieties = 330
Exercises = 342
18. Localization = 347
Tangent space of a variety at a point = 348
Regular and singular points = 351
Coincidence of the geometric and the algebraic dimension of a variety = 352
Local ring of a variety at a point = 353
Localization of a commutative ring at a prime ideal = 355
Local rings = 360
Exercises = 361
19. Factorization of ideals = 365
Kummer's idea to overcome non-unique factorization = 365
Dedekind domains = 367
Fractional ideals = 367
Calculus of factional ideals = 368
Characterization of Dedekind domains = 373
Examples of Dedekind domains = 373
Divisibility of ideals in a Dedekind domain = 374
Exercises = 377
20. Introduction to Galois theory : Solving polynomial equations = 379
Quadratic formula = 379
Cardano's formula for cubic equations = 380
Ferrari's formula for quartic equations = 382
Exercises = 384
21. The Galois group of a field extension = 388
Galois group of a field extension = 388
Examples = 388
Fixed fields = 390
Correspondence between intermediate fields of (L : K) and subgroups of GK L = 391
Quantitative aspects of the Galois correspondence = 394
Dedekind's theorem = 395
Artin's theorem = 397
Galois extensions = 399
Exercises = 401
22. Algebraic Galois extensions = 404
Characterization of algebraic Galois extensions = 404
Separable and inseparable closure in a normal field extension = 405
Main theorem of Galois theory = 407
Examples = 408
Galois groups of finite fields = 413
Primitive element theorem = 414
Constructions with a ruler and a compass = 415
Normal bases = 416
Compositum of Galois extensions = 419
Infinite algebraic Galois extensions = 420
Topological characterization of closed subgroups of a Galois group = 425
Exercises = 429
23. The Galois group of a polynomial = 435
The Galois group of a polynomial as a group of permutations of its roots = 435
Algorithm to determine the Galois group of a polynoimal = 436
Reducing coefficients modulo a prime ideal = 437
Correspondence between the Galois group of a polynomial and file Galois group of a field extension = 438
Characterization of the irreducibility of a polynoimal by the transitivity of its Galois group = 439
Dedekind's reciprocity theorem = 441
Galois groups of cubic and quartic polynomials = 443
Galois groups of certain polynomials of prime degree = 445
Dedekind's theorem on reduction modulo a prime = 446
New proof of the main theorem on symmetric polynomials = 448
Galois group of the general polynomial = 449
Exercises = 450
24. Roots of unity and cyclotomic polynomials = 455
Roots of unity = 455
Cyclotomic polynomials = 457
Irreducibility of the cyclotomic polynomials over xn -1 = 460
Galois group of xn -1 = 460
Examples = 461
Gaussian periods = 463
Kummer's lemma = 465
Wedderburn's theorem = 467
Constructibility of regular polygons = 468
New proof of the quadratic reciprocity law = 470
Exercises = 472
25. Pure equations and cyclic extensions = 476
Pure polynomials yield cyclic Galois groups = 476
Theorem 90 of Hilbert = 477
Cyclic Galois groups yield pure polynomials = 478
Lagrange resolvents = 479
Artin-Schreier theorem = 480
Irreducibility of pure polynomials = 481
Kummer correspondence = 484
Exercises = 487
26. Solvable equations and radical extensions = 490
Solvability by radicals = 490
Radical extensions and their properties = 490
Strong radical extensions = 493
Role of the roots of unity = 494
Expressing the solvability of a polynomial equation by the solvability of a given polynomial equation = 496
Examples = 499
Algorithm to determine the solvability of a given polynomial equation = 501
Examples : Quadratic, cubic and quartic equations = 503
Exercises = 512
27. Epilogue : The idea of Lie theory as a Galois theory for differential equations = 515
Bibliography = 524
Index = 526
