1st Edition

Abstract Algebra A Gentle Introduction

By Gary L. Mullen, James A. Sellers Copyright 2017
    214 Pages
    by CRC Press

    214 Pages
    by Chapman & Hall

     

    Abstract Algebra: A Gentle Introduction advantages a trend in mathematics textbook publishing towards smaller, less expensive and brief introductions to primary courses. The authors move away from the ‘everything for everyone’ approach so common in textbooks. Instead, they provide the reader with coverage of numerous algebraic topics to cover the most important areas of abstract algebra.





    Through a careful selection of topics, supported by interesting applications, the authors Intend the book to be used for a one-semester course in abstract algebra. It is suitable for an introductory course in for mathematics majors. The text is also very suitable for education majors



    who need to have an introduction to the topic.





    As textbooks go through various editions and authors employ the suggestions of numerous well-intentioned reviewers, these book become larger and larger and subsequently more expensive. This book is meant to counter that process. Here students are given a "gentle introduction," meant to provide enough for a course, yet also enough to encourage them toward future study of the topic.




    Features





    • Groups before rings approach


    • Interesting modern applications


    • Appendix includes mathematical induction, the well-ordering principle, sets, functions, permutations, matrices, and complex nubers.


    • Numerous exercises at the end of each section


    • Chapter "Hint and Partial Solutions" offers built in solutions manual




    Elementary Number Theory



    Divisibility 



    Primes and factorization 



    Congruences



    Solving congruences 



    Theorems of Fermat and Euler



    RSA cryptosystem 



    Groups



    De nition of a group 



    Examples of groups



    Subgroups



    Cosets and Lagrange's Theorem



    Rings



    Defiition of a ring 



    Subrings and ideals



    Ring homomorphisms



    Integral domains



    Fields



    Definition and basic properties of a field



    Finite Fields



    Number of elements in a finite field



    How to construct finite fields



    Properties of finite fields



    Polynomials over finite fields



    Permutation polynomials



    Applications



    Orthogonal latin squares



    Die/Hellman key exchange



    Vector Spaces



    Definition and examples



    Basic properties of vector spaces



    Subspaces



    Polynomials



    Basics



    Unique factorization



    Polynomials over the real and complex numbers



    Root formulas



    Linear Codes



    Basics



    Hamming codes



    Encoding



    Decoding



    Further study



    Exercises



    Appendix



    Mathematical induction



    Well-ordering Principle



    Sets 



    Functions 



    Permutations 



    Matrices



    Complex numbers



    Hints and Partial Solutions to Selected Exercises

    Biography

    Gary Mullen is Professor of Mathematics, The Pennsylvania State University, where he earned his Ph.D. His main interest is finite fields, and is founder of the journal "Finite Fields and Their Introduction." He is also the Editor of The Handbook of Finite Fields published by CRC Press.



    James Sellers is Professor and Associate Head for Undergraduate Mathematics, The Pennsylvania State University, where he also earned his Ph.D. He has published many research articles and won awards related to his efforts to advance mathematics education.