## Modern algebra : an introduction

John R. Durbin

Engineers and computer scientists who need a basic understanding of algebra will benefit from this accessible book. The sixth edition includes many carefully worked examples and proofs to guide them through abstract algebra successfully. It introduces the most important kinds of algebraic structures, and helps them improve their ability to understand and work with abstract ideas. New and revised exercise sets are integrated throughout the first four chapters. A more in-depth discussion is also included on Galois Theory. The first six chapters provide engineers and computer scientists with the core of the subject and then the book explores the concepts in more detail.

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[目次]

• Introduction I. Mappings and Operations 1 Mappings 2 Composition 3Operations 4 Composition as an Operation II. Introduction to Groups 5 Definition and Examples 6 Permutations 7 Subgroups 8 Groups and Symmetry III. Equivalence. Congruence. Divisibility 9 Equivalence Relations 10 Congruence. The Division Algorithm 11 Integers Modulo n 12 Greatest Common Divisors. The Euclidean Algorithm 13 Factorization. Euler's Phi-Function IV. Groups 14 Elementary Properties 15 Generators. Direct Products 16 Cosets 17 Lagrange's Theorem. Cyclic Groups 18 Isomorphism 19 More on Isomorphism 20 Cayley's Theorem Appendix: RSA Algorithm V. Group Homomorphisms 21 Homomorphisms of Groups. Kernels 22 Quotient Groups 23 The Fundamental Homomorphism Theorm VI. Introduction to Rings 24 Definition and Examples 25 Integral Domains. Subrings 26 Fields 27 Isomorphism. Characteristic VII. The Familiar Number Systems 28 Ordered Integral Domains 29 The Integers 30 Field of Quotients. The Field of Rational Numbers 31 Ordered Fields. The Field of Real Numbers 32 The Field of Complex Numbers 33 Complex Roots of Unity VIII. Polynomials 34 Definition and Elementary Properties 35 The Division Algorithm 36 Factorization of Polynomials 37 Unique Factorization Domains IX. Quotient Rings 38 Homomorphisms of Rings. Ideals 39 Quotient Rings 40 Quotient Rings of F[X] 41 Factorization and Ideals X. Galois Theory: Overview 42 Simple Extensions. Degree 43 Roots of Polynomials 44 Fundamental Theorem: Introduction XI. Galois Theory 45 Galois Groups 46 Splitting Fields. Galois Groups 47 Separability and Normality 48 Fundamental Theorem of Galois Theory 49 Solvability by Radicals 50 Finite Fields XII. Geometric Constructions 51 Three Famous Problems 52 Constructible Numbers 53 Impossible Constructions XIII. Solvable and Alternating Groups 54 Isomorphis Theorems. Solvable Groups 55 Alternation Groups XIV. Applications of Permutation Groups 56 Groups Acting on Sets 57 Burnside's Counting Theorem 58 Sylow's Theorem XV. Symmetry 59 Finite Symmetry Groups 60 Infinite Two-dimensional Symmetry Groups 61 On Crystallographic Groups 62 The Euclidean Group XVI. Lattices and Boolean Algebras 63 Partially Ordered Sets 64 Lattices 65 Boolean Algebras 66 Finite Boolean Algebras A. Sets B. Proofs C. Mathematical Induction D. Linear Algebra E. Solutions to Selected Problems Photo Credit List Index of Notation Index

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書名 Modern algebra : an introduction Durbin, John R. Wiley c2009 6th ed xiii, 335 p. 26 cm 9780470384435 BA89568259 ※クリックでCiNii Booksを表示 英語 アメリカ合衆国
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