Abstract Algebra with Applications: Volume 1: Vector Spaces and Groups

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CRC Press, Oct 18, 1993 - Mathematics - 776 pages
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A comprehensive presentation of abstract algebra and an in-depth treatment of the applications of algebraic techniques and the relationship of algebra to other disciplines, such as number theory, combinatorics, geometry, topology, differential equations, and Markov chains.
 

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lagrange identity

Contents

VECTOR SPACES
1
Examples from geometry I Vectors as equivalence classes of arrows 2 Addition
19
Examples from electrical engineering and economics 22 Fields 24 Matrices 25
42
Vector spaces
49
Polynomial and power series rings
54
Vector spaces and subspaces 49 Examples 50 Free and generating sets 54 Bases
55
Dimension 58 Dimension formula for subspaces 58 Direct decompositions 59 Quo
65
Polynomials in one or more variables 54 Division with remainder 57 Roots and their
67
Pure inseparability 275 Decomposition of an algebraic field extension into a separable
278
Field theory and integral ring extensions
291
Scalar products 273 Orthogonality 276 Orthogonal decompositions 279 Orthonor
296
Affine algebras 301 Noethers normalization theorem 302 Zariskis lemma 304
304
Ring theory and algebraic geometry
318
Adjoint of a linear mapping 303 Properties 304 Trace and operator norm 306 Nor
324
Bilinear forms
331
Affine varieties 318 Correspondence between varieties and ideals 322 Hilberts Null
342

Linear and affine mappings
72
Ideals 76 Ideals generated by a set 77 Principal ideals 77 Simple rings 78 Quotient
86
phism theorem 82 Dimension formula for linear mappings 83 Vector spaces of linear
88
Abstract affine geometry
95
Affine spaces and subspaces 95 Dimension formula for affine subspaces 99 Affine
105
Primary decompositions 102 Symbolic powers 105 Exercises
107
Representation of linear mappings by matrices
111
Parameterizations basis isomorphisms and coordinate transformations 111 Matrix
119
principal ideal domains 119 Euclidean domains 119 Examples 120 Euclidean algo
128
to equivalence 122 Similarity of square matrices 124 Canonical forms for projections
130
Determinants
138
Determinants as natural constructions in elimination theory 138 Existence and unique
154
Transmission of the unique factorization property from R to RXI xn 135 Unique
155
product by minors of the individual factors 165 Determinant of an endomorphism as
173
Quadratic reciprocity law 166 Three theorems of Fermat 169 RamanujanNagell the
178
Gershgorins theorem 183 Geometric and algebraic multiplicity 185 Diagonalizability
186
Modules 185 Free modules 185 Submodules and quotient modules 186 Module
196
Classification of endomorphisms up to similarity
205
Cohens theorem 205 Noetherian induction 208 Primary decompositions in Noethe
215
Field extensions
216
Determinantal and elementary divisors 219 Exercises
223
Characterization of simple extensions 227 Algebraically generating and independent
238
Complexification 230 Tensor product of vector spaces 231 Properties 232 Basefield
245
Adjoining roots of polynomials 248 Splitting fields 250 Normal extensions 250
262
Oriented areas and volumes 255 Vector product 258 Applications in geometry
268
of alternating forms 341 Pfaffian 342 Classification of real symmetric forms by rank
358
Tangent space of a variety at a point 348 Regular and singular points 351 Coinci
361
Factorization of ideals
365
Solving polynomial equations
379
Grouptheoretical interpretation of the Gaussian algorithm 373 Gaussian decompo
384
Markov processes 393 Transition matrices and graphs 393 Examples 394 Longterm
400
Galois group of a field extension 388 Examples 388 Fixed fields 390 Correspondence
401
405 Cyclic classes 407 Mean first passage matrix 411 Examples in genetics 413
413
Galois groups of finite fields 413 Primitive element theorem 414 Constructions with
429
Examples from electrical engineering and economics 424 Existence and uniqueness
444
Galois groups of cubic and quartic polynomials 443 Galois groups of certain polyno
450
Symmetries in art and nature 451 Symmetry transformations 453 Types of one
460
Roots of unity 455 Cyclotomic polynomials 457 Irreducibility of the cyclotomic poly
463
Subgroups and cosets
476
Lagranges theorem 487 Applications in number theory 488 Exercises
489
Cycle decomposition 498 Cyclestructure and conjugacy 498 Generating sets
505
Exercises
523
groups 536 Semidirect products 537 Exercises
541
Free groups 549 Universal property 550 Generators and relations 553 Von Dycks
557
Torsion subgroup of an abelian group 557 Classification of finitely generated abelian
570
Group actions 575 Examples 575 Orbits and stabilizers 577 Transitivity 577
591
Actions of groups on groups and coset spaces 596 Class equation 597 Deriving struc
604
of finite nilpotent groups 617 Affine subgroups of Symn 617 Characterization of
622
Topological groups and their subgroups 624 Continuous homomorphisms 628 Iso
641
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