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First Course in Abstract Algebra: with Applications (3rd Edition) by Joseph J. Rotman

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First Course in Abstract Algebra: with Applications (3rd Edition) written by Joseph J. Rotman , University of Illinois, at Urbana-Champaign. This text introduces readers to the algebraic concepts of group and rings, providing a comprehensive discussion of theory as well as a significant number of applications for each. Number Theory: Induction; Binomial Coefficients; Greatest Common Divisors; The Fundamental Theorem of Arithmetic Congruences; Dates and Days. Groups I: Some Set Theory; Permutations; Groups; Subgroups and Lagrange's Theorem; Homomorphisms; Quotient Groups; Group Actions; Counting with Groups. Commutative Rings I: First Properties; Fields; Polynomials; Homomorphisms; Greatest Common Divisors; Unique Factorization; Irreducibility; Quotient Rings and Finite Fields; Officers, Magic, Fertilizer, and Horizons. Linear Algebra: Vector Spaces; Euclidean Constructions; Linear Transformations; Determinants; Codes; Canonical Forms. Fields: Classical Formulas; Insolvability of the General Quintic; Epilog. Groups II: Finite Abelian Groups; The Sylow Theorems; Ornamental Symmetry. Commutative Rings III: Prime Ideals and Maximal Ideals; Unique Factorization; Noetherian Rings; Varieties; Grobner Bases.

First Course in Abstract Algebra: with Applications (3rd Edition) written by Joseph J. Rotman cover the following topics.

  • 1. Number Theory
    1.1 Induction
    1.2 Binomial Coefficients
    1.3 Greatest Common Divisors
    1.4 The Fundamental Theorem of Arithmetic
    1.5 Congruences
    1.6 Dates and Days

  • 2. Groups I
    2.1 Some Set Theory, Functions, Equivalence Relations
    2.2 Permutations
    2.3 Groups, Symmetry
    2.4 Subgroups and Lagrange’s Theorem
    2.5 Homomorphisms
    2.6 Quotient Groups
    2.7 Group Actions
    2.8 Counting with Groups

  • 3. Commutative Rings I
    3.1 First Properties
    3.2 Fields
    3.3 Polynomials
    3.4 Homomorphisms
    3.5 Greatest Common Divisors, Euclidean Rings
    3.6 Unique Factorization
    3.7 Irreducibility
    3.8 Quotient Rings and Finite Fields
    3.9 Officers, Magic, Fertilizer, and Horizons, Officers, Magic, Fertilizer, Horizons

  • 4. Linear Algebra
    4.1 Vector Spaces, Gaussian Elimination
    4.2 Euclidean Constructions
    4.3 Linear Transformations
    4.4 Determinants
    4.5 Codes, Block Codes, Linear Codes

  • 5 Fields
    5.1 Classical Formulas, Vi`ete’s Cubic Formula
    5.2 Insolvability of the General Quintic, Formulas and Solvability by Radicals, Translation into Group Theory
    5.3 Epilog

  • 6. Groups II
    6.1 Finite Abelian Groups
    6.2 The Sylow Theorems
    6.3 Ornamental Symmetry

  • 7. Commutative Rings II
    7.1 Prime Ideals and Maximal Ideals
    7.2 Unique Factorization
    7.3 Noetherian Rings
    7.4 Varieties
    7.5 Gr¨obner Bases, Monomial Orders, Generalized Division Algorithm, Gr¨obner Bases

  • Appendix
    A. Inequalities
    B. Pseudocodes
    Hints for Selected Exercises

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