Description
Apple Academic Press Inc. A First Course In Abstract Algebrarings Groups And Fields Third Edition 2014 Edition by Marlow Anderson, Todd Feil
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 EditionMakes 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 theoryThe 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. Numbers, Polynomials, and Factoring The Natural Numbers The Integers Modular Arithmetic Polynomials with Rational CoefficientsFactorization of PolynomialsSection I in a NutshellRings, Domains, and Fields Rings Subrings and Unity Integral Domains and Fields Ideals Polynomials over a Field Section II in a NutshellRing 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 NutshellGroups Symmetries of Geometric Figures PermutationsAbstract Groups Subgroups Cyclic Groups Section IV in a NutshellGroup Homomorphisms Group Homomorphisms Structure and Representation Cosets and Lagrange's Theorem Groups of CosetsThe Isomorphism Theorem for Groups Section V in a NutshellTopics from Group Theory The Alternating Groups Sylow Theory: The Preliminaries Sylow Theory: The Theorems Solvable Groups Section VI in a NutshellUnique Factorization Quadratic Extensions of the Integers FactorizationUnique Factorization Polynomials with Integer Coefficients Euclidean Domains Section VII in a NutshellConstructibility Problems Constructions with Compass and Straightedge Constructibility and Quadratic Field Extensions The Impossibility of Certain Constructions Section VIII in a NutshellVector 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 NutshellGalois 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