## A first course in abstract algebra : rings, groups, and fields

Marlow Anderson, Todd Feil

Most abstract algebra texts begin with groups, then proceed to rings and fields. While groups are the logically simplest of the structures, the motivation for studying groups can be somewhat lost on students approaching abstract algebra for the first time. To engage and motivate them, starting with something students know and abstracting from there is more natural-and ultimately more effective. Authors Anderson and Feil developed A First Course in Abstract Algebra: Rings, Groups and Fields based upon that conviction. The text begins with ring theory, building upon students' familiarity with integers and polynomials. Later, when students have become more experienced, it introduces groups. The last section of the book develops Galois Theory with the goal of showing the impossibility of solving the quintic with radicals. Each section of the book ends with a "Section in a Nutshell" synopsis of important definitions and theorems. Each chapter includes "Quick Exercises" that reinforce the topic addressed and are designed to be worked as the text is read. Problem sets at the end of each chapter begin with "Warm-Up Exercises" that test fundamental comprehension, followed by regular exercises, both computational and "supply the proof" problems. A Hints and Answers section is provided at the end of the book. As stated in the title, this book is designed for a first course--either one or two semesters in abstract algebra. It requires only a typical calculus sequence as a prerequisite and does not assume any familiarity with linear algebra or complex numbers.

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[目次]

• 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 Polynomials over a Field Section II in a Nutshell UNIQUE FACTORIZATION Associates and Irreducibles Factorization and Ideals Principal Ideal Domains Primes and Unique Factorization Polynomials with Integer Coefficients Euclidean Domains Section III 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 IV in a Nutshell GROUPS Symmetries of Figures in the Plane Symmetries of Figures in Space Abstract Groups Subgroups Cyclic Groups Section V in a Nutshell GROUP HOMOMORPHISMS AND PERMUTATIONS Group Homomorphisms Group Isomorphisms Permutations and Cayley's Theorem More About Permutations Cosets and Lagrange's Theorem Groups of Cosets The Isomorphism Theorem for Groups The Alternating Groups Fundamental Theorem for Finite Abelian Groups Solvable Groups Section VI in a Nutshell CONSTRUCTIBILITY PROBLEMS Constructions with Compass and Straightedge Constructibility and Quadratic Field Extensions The Impossibility of Certain Constructions Section VII 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 VIII in a Nutshell GALOIS THEORY The Splitting Field Finite Fields Galois Groups The Fundamental Theorem of Galois Theory Solving Polynomials by Radicals Section IX in a Nutshell Hints and Solutions Guide to Notation Index

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書名 A first course in abstract algebra : rings, groups, and fields Anderson, Marlow Feil, Todd Chapman & Hall/CRC c2005 2nd ed xviii, 673 p. 25 cm 1584885157 BA71947756 ※クリックでCiNii Booksを表示 英語 アメリカ合衆国
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