A First Course in Abstract Algebra - Anderson, Marlow

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Résumé :
Like its popular predecessors, this text develops ring theory first by drawing on students' familiarity with integers and polynomials. This unique approach motivates students in studying abstract algebra and helps them understand the power of abstraction. This edition makes it easier to teach unique factorization as an optional topic and reorganizes the core material on rings, integral domains, and fields. Along with new exercises on Galois theory, it also includes a more detailed treatment of permutations as well as new chapters on Sylow theorems....

Biographie:
Marlow Anderson, Todd Feil

Sommaire:
Numbers, Polynomials, and Factoring The Natural Numbers The Integers Modular Arithmetic Polynomials with Rational Coefficients Factorization of Polynomials Section 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 Homomorphisms The 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 Permutations Abstract Groups Subgroups Cyclic Groups Section IV in a Nutshell Group Homomorphisms Group Homomorphisms Structure and Representation Cosets and Lagrange's Theorem Groups of Cosets The 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 Factorization Unique 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 I Vector 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 Theory Solving Polynomials by Radicals Section X in a Nutshell Hints and Solutions Guide to Notation Index