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A Concrete Introduction to Higher Algebra

  • Textbook
  • © 1995

Overview

Authors:
  1. Lindsay N. Childs

    1. Department of Mathematics, SUNY at Albany, Albany, USA

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Part of the book series: Undergraduate Texts in Mathematics (UTM)

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Table of contents (30 chapters)

  1. Front Matter

    Pages i-xv
    Download chapter PDF

  2. Numbers

    • Lindsay N. Childs
    Pages 1-7

  3. Induction

    • Lindsay N. Childs
    Pages 8-24

  4. Euclid’s Algorithm

    • Lindsay N. Childs
    Pages 25-46

  5. Unique Factorization

    • Lindsay N. Childs
    Pages 47-62

  6. Congruences

    • Lindsay N. Childs
    Pages 63-75

  7. Congruence Classes

    • Lindsay N. Childs
    Pages 76-90

  8. Applications of Congruences

    • Lindsay N. Childs
    Pages 91-117

  9. Rings and Fields

    • Lindsay N. Childs
    Pages 118-133

  10. Fermat’s and Euler’s Theorems

    • Lindsay N. Childs
    Pages 134-154

  11. Applications of Fermat’s and Euler’s Theorems

    • Lindsay N. Childs
    Pages 155-179

  12. On Groups

    • Lindsay N. Childs
    Pages 180-193

  13. The Chinese Remainder Theorem

    • Lindsay N. Childs
    Pages 194-207

  14. Matrices and Codes

    • Lindsay N. Childs
    Pages 208-230

  15. Polynomials

    • Lindsay N. Childs
    Pages 231-238

  16. Unique Factorization

    • Lindsay N. Childs
    Pages 239-252

  17. The Fundamental Theorem of Algebra

    • Lindsay N. Childs
    Pages 253-276

  18. Derivatives

    • Lindsay N. Childs
    Pages 277-285

  19. Factoring in ℚ[x], I

    • Lindsay N. Childs
    Pages 286-292

  20. The Binomial Theorem in Characteristic p

    • Lindsay N. Childs
    Pages 293-301
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Keywords

About this book

This book is written as an introduction to higher algebra for students with a background of a year of calculus. The first edition of this book emerged from a set of notes written in the 1970sfor a sophomore-junior level course at the University at Albany entitled "Classical Algebra." The objective of the course, and the book, is to give students enough experience in the algebraic theory of the integers and polynomials to appre­ ciate the basic concepts of abstract algebra. The main theoretical thread is to develop algebraic properties of the ring of integers: unique factorization into primes, congruences and congruence classes, Fermat's theorem, the Chinese remainder theorem; and then again for the ring of polynomials. Doing so leads to the study of simple field extensions, and, in particular, to an exposition of finite fields. Elementary properties of rings, fields, groups, and homomorphisms of these objects are introduced and used as needed in the development. Concurrently with the theoretical development, the book presents a broad variety of applications, to cryptography, error-correcting codes, Latin squares, tournaments, techniques of integration, and especially to elemen­ tary and computational number theory. A student who asks, "Why am I learning this?," willfind answers usually within a chapter or two. For a first course in algebra, the book offers a couple of advantages. • By building the algebra out of numbers and polynomials, the book takes maximal advantage of the student's prior experience in algebra and arithmetic. New concepts arise in a familiar context.

Reviews

From the reviews:

"The user-friendly exposition is appropriate for the intended audience. Exercises often appear in the text at the point they are relevant, as well as at the end of the section or chapter. Hints for selected exercises are given at the end of the book. There is sufficient material for a two-semester course and various suggestions for one-semester courses are provided. Although the overall organization remains the same in the second edition¿Changes include the following: greater emphasis on finite groups, more explicit use of homomorphisms, increased use of the Chinese remainder theorem, coverage of cubic and quartic polynomial equations, and applications which use the discrete Fourier transform." MATHEMATICAL REVIEWS

Authors and Affiliations

  • Department of Mathematics, SUNY at Albany, Albany, USA

    Lindsay N. Childs

Bibliographic Information

  • Book Title: A Concrete Introduction to Higher Algebra

  • Authors: Lindsay N. Childs

  • Series Title: Undergraduate Texts in Mathematics

  • DOI: https://doi.org/10.1007/978-1-4419-8702-0

  • Publisher: Springer New York, NY

  • eBook Packages: Springer Book Archive

  • Copyright Information: Springer Science+Business Media New York 1995

  • Softcover ISBN: 978-0-387-98999-0Published: 14 January 2000

  • eBook ISBN: 978-1-4419-8702-0Published: 04 December 2012

  • Series ISSN: 0172-6056

  • Series E-ISSN: 2197-5604

  • Edition Number: 2

  • Number of Pages: XV, 522

  • Topics: Algebra

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