Algebra (Cambridge Mathematical Textbooks)
- Length: 488 pages
- Edition: 1
- Language: English
- Publisher: Cambridge University Press
- Publication Date: 2021-07-29
- ISBN-10: 1108958230
- ISBN-13: 9781108958233
- Sales Rank: #160944 (See Top 100 Books)
From rings to modules to groups to fields, this undergraduate introduction to abstract algebra follows an unconventional path. The text emphasizes a modern perspective on the subject, with gentle mentions of the unifying categorical principles underlying the various constructions and the role of universal properties. A key feature is the treatment of modules, including a proof of the classification theorem for finitely generated modules over Euclidean domains. Noetherian modules and some of the language of exact complexes are introduced. In addition, standard topics – such as the Chinese Remainder Theorem, the Gauss Lemma, the Sylow Theorems, simplicity of alternating groups, standard results on field extensions, and the Fundamental Theorem of Galois Theory – are all treated in detail. Students will appreciate the text’s conversational style, 400+ exercises, an appendix with complete solutions to around 150 of the main text problems, and an appendix with general background on basic logic and naïve set theory.
Cover Half-title Series information Title page Copyright information Contents Introduction Before we Begin Part I Rings 1 The Integers 1.1 The Well-Ordering Principle and Induction 1.2 ‘Division with Remainder’ in Z 1.3 Greatest Common Divisors 1.4 The Fundamental Theorem of Arithmetic 2 Modular Arithmetic 2.1 Equivalence Relations and Quotients 2.2 Congruence mod n 2.3 Algebra in Z/nZ 2.4 Properties of the Operations +, · on Z/nZ 2.5 Fermat’s Little Theorem, and the RSA Encryption System 3 Rings 3.1 Definition and Examples 3.2 Basic Properties 3.3 Special Types of Rings 4 The Category of Rings 4.1 Cartesian Products 4.2 Subrings 4.3 Ring Homomorphisms 4.4 Isomorphisms of Rings 5 Canonical Decomposition, Quotients, and Isomorphism Theorems 5.1 Rings: Canonical Decomposition, I 5.2 Kernels and Ideals 5.3 Quotient Rings 5.4 Rings: Canonical Decomposition, II 5.5 The First Isomorphism Theorem 5.6 The Chinese Remainder Theorem 5.7 The Third Isomorphism Theorem 6 Integral Domains 6.1 Prime and Maximal Ideals 6.2 Primes and Irreducibles 6.3 Euclidean Domains and PIDs 6.4 PIDs and UFDs 6.5 The Field of Fractions of an Integral Domain 7 Polynomial Rings and Factorization 7.1 Fermat’s Last Theorem for Polynomials 7.2 The Polynomial Ring with Coefficients in a Field 7.3 Irreducibility in Polynomial Rings 7.4 Irreducibility in Q[x] and Z[x] 7.5 Irreducibility Tests in Z[x] Part II Modules 8 Modules and Abelian Groups 8.1 Vector Spaces and Ideals, Revisited 8.2 The Category of R-Modules 8.3 Submodules, Direct Sums 8.4 Canonical Decomposition and Quotients 8.5 Isomorphism Theorems 9 Modules over Integral Domains 9.1 Free Modules 9.2 Modules from Matrices 9.3 Finitely Generated vs. Finitely Presented 9.4 Vector Spaces are Free Modules 9.5 Finitely Generated Modules over Euclidean Domains 9.6 Linear Transformations and Modules over k[t] 10 Abelian Groups 10.1 The Category of Abelian Groups 10.2 Cyclic Groups, and Orders of Elements 10.3 The Classification Theorem 10.4 Fermat’s Theorem on Sums of Squares Part III Groups 11 Groups—Preliminaries 11.1 Groups and their Category 11.2 Why Groups? Actions of a Group 11.3 Cyclic, Dihedral, Symmetric, Free Groups 11.4 Canonical Decomposition, Normality, and Quotients 11.5 Isomorphism Theorems 12 Basic Results on Finite Groups 12.1 The Index of a Subgroup, and Lagrange’s Theorem 12.2 Stabilizers and the Class Equation 12.3 Classification and Simplicity 12.4 Sylow’s Theorems: Statements, Applications 12.5 Sylow’s Theorems: Proofs 12.6 Simplicity of A[sub(n)] 12.7 Solvable Groups Part IV Fields 13 Field Extensions 13.1 Fields and Homomorphisms of Fields 13.2 Finite Extensions and the Degree of an Extension 13.3 Simple Extensions 13.4 Algebraic Extensions 13.5 Application: ‘Geometric Impossibilities’ 14 Normal and Separable Extensions, and Splitting Fields 14.1 Simple Extensions, Again 14.2 Splitting Fields 14.3 Normal Extensions 14.4 Separable Extensions; and Simple Extensions Once Again 14.5 Application: Finite Fields 15 Galois Theory 15.1 Galois Groups and Galois Extensions 15.2 Characterization of Galois Extensions 15.3 The Fundamental Theorem of Galois Theory 15.4 Galois Groups of Polynomials 15.5 Solving Polynomial Equations by Radicals 15.6 Other Applications Appendix A Background A.1 Sets—Basic Notation A.2 Logic A.3 Quantifiers A.4 Types of Proof: Direct, Contradiction, Contrapositive A.5 Set Operations A.6 Functions A.7 Injective/Surjective/Bijective Functions A.8 Relations; Equivalence Relations and Quotient Sets A.9 Universal Property of Quotients and Canonical Decomposition Appendix B Solutions to Selected Exercises B.1 The Integers B.2 Modular Arithmetic B.3 Rings B.4 The Category of Rings B.5 Canonical Decomposition, Quotients, and Isomorphism Theorems B.6 Integral Domains B.7 Polynomial Rings and Factorization B.8 Modules and Abelian Groups B.9 Modules over Integral Domains B.10 Abelian Groups B.11 Groups—Preliminaries B.12 Basic Results on Finite Groups B.13 Field Extensions B.14 Normal and Separable Extensions, and Splitting Fields B.15 Galois Theory Index of Definitions Index of Theorems Subject Index
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