Descripciones mejoradas de Syndetics:

This book blends the theoretical with the practical in the instruction of modern algebra. Modern algebra is usually taught from the point of view of its intrinsic interest, without using applications. Many readers lose interest when they do not see the relevance of the subject and often become skeptical of the explanation that the material will be used later. The author believes by providing interesting and nontrivial applications, the student will better appreciate and understand the subject.

Tabla de contenidos provista por Syndetics

• 1. Introduction(p. 1)
• Classical Algebra(p. 1)
• Modern Algebra(p. 2)
• Binary Operations(p. 2)
• Algebraic Structures(p. 5)
• Extending Number Systems(p. 6)
• 2. Boolean Algebras(p. 8)
• Algebra of Sets(p. 8)
• Number of Elements in a Set(p. 13)
• Boolean Algebras(p. 15)
• Switching Circuits(p. 23)
• Posets and Lattices(p. 25)
• Normal Forms and Simplification of Circuits(p. 28)
• Transistor Gates(p. 39)
• Representation Theorem(p. 42)
• Exercises(p. 44)
• 3. Groups(p. 51)
• Groups and Symmetries(p. 52)
• Subgroups(p. 58)
• Cyclic Groups and Dihedral Groups(p. 60)
• Morphisms(p. 64)
• Permutation Groups(p. 67)
• Even and Odd Permutations(p. 73)
• Cayley's Representation Theorem(p. 77)
• Exercises(p. 78)
• 4. Quotient Groups(p. 82)
• Equivalence Relations(p. 82)
• Cosets and Lagrange's Theorem(p. 85)
• Normal Subgroups and Quotient Groups(p. 88)
• Morphism Theorem(p. 93)
• Direct Products(p. 98)
• Groups of Low Order(p. 102)
• Action of a Group on a Set(p. 104)
• Exercises(p. 108)
• 5. Symmetry Groups in Three Dimensions(p. 112)
• Translations and the Euclidean Group(p. 112)
• Matrix Groups(p. 115)
• Finite Groups in Two Dimensions(p. 117)
• Proper Rotations of Regular Solids(p. 119)
• Finite Rotation Groups in Three Dimensions(p. 124)
• Crystallographic Groups(p. 129)
• Exercises(p. 130)
• 6. Polya-Burnside Method of Enumeration(p. 133)
• Burnside's Theorem(p. 134)
• Necklace Problems(p. 135)
• Coloring Polyhedra(p. 138)
• Counting Switching Circuits(p. 140)
• Exercises(p. 145)
• 7. Monoids and Machines(p. 147)
• Monoids and Semigroups(p. 147)
• Finite-State Machines(p. 152)
• Quotient Monoids and the Monoid of a Machine(p. 155)
• Exercises(p. 160)
• 8. Rings and Fields(p. 166)
• Rings(p. 166)
• Integral Domains and Fields(p. 171)
• Subrings and Morphisms of Rings(p. 173)
• New Rings from Old(p. 176)
• Field of Fractions(p. 182)
• Convolution Fractions(p. 184)
• Exercises(p. 188)
• 9. Polynomial and Euclidean Rings(p. 193)
• Division Algorithm(p. 193)
• Euclidean Algorithm(p. 198)
• Unique Factorization(p. 202)
• Factoring Real and Complex Polynomials(p. 205)
• Factoring Rational and Integral Polynomials(p. 207)
• Factoring Polynomials over Finite Fields(p. 211)
• Linear Congruences and the Chinese Remainder Theorem(p. 212)
• Exercises(p. 217)
• 10. Quotient Rings(p. 221)
• Ideals and Quotient Rings(p. 221)
• Computations in Quotient Rings(p. 225)
• Morphism Theorem(p. 227)
• Quotient Polynomial Rings that are Fields(p. 228)
• Exercises(p. 232)
• 11. Field Extensions(p. 235)
• Field Extensions(p. 235)
• Algebraic Numbers(p. 238)
• Galois Fields(p. 243)
• Primitive Elements(p. 246)
• Exercises(p. 252)
• 12. Latin Squares(p. 255)
• Latin Squares(p. 255)
• Orthogonal Latin Squares(p. 258)
• Finite Geometries(p. 262)
• Magic Squares(p. 266)
• Exercises(p. 269)
• 13. Geometrical Constructions(p. 272)
• Constructible Numbers(p. 273)
• Duplicating the Cube(p. 278)
• Trisecting an Angle(p. 278)
• Squaring the Circle(p. 281)
• Constructing Regular Polygons(p. 281)
• A Nonconstructible Number of Degree Four(p. 283)
• Exercises(p. 284)
• 14. Error-Correcting Codes(p. 286)
• The Coding Problem(p. 287)
• Simple Codes(p. 290)
• Polynomial Representation(p. 293)
• Matrix Representation(p. 299)
• Error Correcting and Decoding(p. 304)
• BCH Codes(p. 309)
• Exercises(p. 314)
• Bibliography and References(p. 319)
• Answers to the Odd-Numbered Exercises(p. 324)
• Glossary of Symbols(p. 337)
• Index(p. 341)

Notas de autor provistas por Syndetics

William J. Gilbert is Professor in the Department of Pure Mathematics at the University of Waterloo, Ontario, Canada.