Like its popular predecessors, A First Course in Abstract Algebra: Rings, Groups, and Fields, Third Edition develops ring theory first by drawing on students' familiarity with integers and polynomials. This unique approach motivates students in the study of abstract algebra and helps them understand the power of abstraction. The authors introduce groups later on using examples of symmetries of figures in the plane and space as well as permutations. New to the Third Edition Makes it easier to teach unique factorization as an optional topic Reorganizes the core material on rings, integral domains, and fieldsIncludes a more detailed treatment of permutationsIntroduces more topics in group theory, including new chapters on Sylow theoremsProvides many new exercises on Galois theory The text includes straightforward exercises within each chapter for students to quickly verify facts, warm-up exercises following the chapter that test fundamental comprehension, and regular exercises concluding the chapter that consist of computational and supply-the-proof problems. Historical remarks discuss the history of algebra to underscore certain pedagogical points. Each section also provides a synopsis that presents important definitions and theorems, allowing students to verify the major topics from the section.
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Numbers, Polynomials, and Factoring The Natural Numbers The Integers Modular Arithmetic Polynomials with Rational CoefficientsFactorization of PolynomialsSection I in a Nutshell Rings, Domains, and Fields Rings Subrings and Unity Integral Domains and Fields Ideals Polynomials over a Field Section II in a Nutshell Ring Homomorphisms and Ideals Ring HomomorphismsThe Kernel Rings of Cosets The Isomorphism Theorem for Rings Maximal and Prime Ideals The Chinese Remainder Theorem Section III in a Nutshell Groups Symmetries of Geometric Figures PermutationsAbstract Groups Subgroups Cyclic Groups Section IV in a Nutshell Group Homomorphisms Group Homomorphisms Structure and Representation Cosets and Lagrange's Theorem Groups of CosetsThe Isomorphism Theorem for Groups Section V in a Nutshell Topics from Group Theory The Alternating Groups Sylow Theory: The Preliminaries Sylow Theory: The Theorems Solvable Groups Section VI in a Nutshell Unique Factorization Quadratic Extensions of the Integers FactorizationUnique Factorization Polynomials with Integer Coefficients Euclidean Domains Section VII in a Nutshell Constructibility Problems Constructions with Compass and Straightedge Constructibility and Quadratic Field Extensions The Impossibility of Certain Constructions Section VIII in a Nutshell Vector Spaces and Field Extensions Vector Spaces IVector Spaces II Field Extensions and Kronecker's Theorem Algebraic Field Extensions Finite Extensions and Constructibility Revisited Section IX in a Nutshell Galois Theory The Splitting Field Finite Fields Galois Groups The Fundamental Theorem of Galois TheorySolving Polynomials by Radicals Section X in a Nutshell Hints and Solutions Guide to Notation Index
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"I am a fan of the rings-first approach to algebra, agreeing with the authors that students' familiarity with the integers and with polynomials renders rings more intuitive and accessible than groups. But this book has many other virtues besides presenting the material in this order. For example, each section is preceded and followed by short sections that try to put the material into a broader context. ... This is definitely a book worth considering for textbook adoption."-MAA Reviews, November 2014 Praise for the Second Edition:"I was quickly won over by the book ... . The book is very complete, containing more than enough material for a two semester course in undergraduate abstract algebra ... . Even though there was a great deal of material presented, I found the book to be very well organized. ... There are a lot of things that I like about this book. ... [It is] well written and will help students to see the big picture. ... All in all it seems that a lot of thought went into this book, resulting in a comprehensive, well-written, readable book for undergraduates first learning abstract algebra."-MAA Online "A remarkable feature of the book is that it starts first with the concept of a ring, while groups are introduced later. The reason of that is that students are usually more familiar with various number domains rather than the mappings and matrices. There is a huge number of examples in the book ... . The book contains a lot of nice exercises of various degrees of difficulty so that it can also be used as a practice book."-EMS Newsletter, March 2006
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3. utgave
Apple Academic Press Inc.
1270 gr
254 mm
178 mm
05, U
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