000 | 00868camuuu200253 a 4500 | |

001 | 000000220226 | |

005 | 19970428165745.0 | |

008 | 930812s1994 nyua b 001 0 eng | |

010 | ▼a 93032090 //r94 | |

020 | ▼a 0824791444 (v. 1 : acid-free paper) | |

020 | ▼a 0824791592 (v. 2 : acid-free paper) | |

040 | ▼a DLC ▼c DLC | |

049 | ▼l 111058417 ▼v 1 ▼l 111058418 ▼v 2 ▼l 121025459 ▼v 1 | |

050 | 0 0 | ▼a QA162 ▼b .S66 1994 |

082 | 0 0 | ▼a 512/.02 ▼2 20 |

090 | ▼a 512.02 ▼b S757a | |

100 | 1 | ▼a Spindler, Karlheinz , ▼d 1960-. |

245 | 1 0 | ▼a Abstract algebra with applications: ▼b in two volumes / ▼c Karlheinz Spindler. |

260 | ▼a New York : ▼b M. Dekker , ▼c c1994. | |

300 | ▼a 2 v. : ▼b ill. ; ▼c 26 cm. | |

504 | ▼a Includes bibliographical references and indexes. | |

505 | 0 | ▼a v. 1. Vector spaces and groups -- v. 2. Rings and fields. |

650 | 0 | ▼a Algebra, Abstract. |

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### Contents information

#### Table of Contents

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