A Concrete Introduction to Higher Algebra / Edition 3

by Lindsay N. Childs

Overview

This book is written as an introduction to higher algebra for students with a background of a year of calculus. It has been used as a successful undergraduate text for more than 20 years. The objective of the book is to give students enough experience in the algebraic theory of the integers and polynomials to appreciate the basic concepts of abstract algebra. New sections on Luhn's formula, Cosets and equations, and detaching coefficients are included. The later chapters include expositions of Rabiin's probabilistic primality test, quadratic reciprocity, and the classification of finite fields. Over 900 exercises, ranging from routine examples to extensions of theory, are found throughout the book; hints and answers for many of them are included in an appendix.

Product Details

ISBN-13:	9781441925619
Publisher:	Springer New York
Publication date:	11/19/2010
Series:	Undergraduate Texts in Mathematics , #159
Edition description:	Softcover reprint of hardcover 3rd ed. 2009
Pages:	604
Product dimensions:	6.10(w) x 9.20(h) x 1.30(d)

Table of Contents

Part I Numbers

1 Numbers 3

2 Induction 9

3 Euclid's Algorithm 27

4 Unique Factorization 53

5 Congruence 71

Part II Congruence classes and rings

6 Congruence Classes 93

7 Rings and Fields 123

8 Matrices and Codes 147

Part III Congruences and Groups

9 Fermat's and Euler's Theorems 171

10 Applications of Eurler's Theorem 201

11 Groups 223

12 The Chinese Remainder Theorem 253

Part IV Polynomials

13 Polynomials 285

14 Unique Factorization 295

15 The Fundamental Theorem of Algebra 307

16 Polynomials in Q[x] 339

17 Congruences and the Chinese Remainder Theorem 355

18 Fast Polynomial Multiplication 373

Part V Primitive Roots

19 Cyclic Groups and Crytography 387

20 Carmichael Numbers 413

21 Quadratic Reciprocity 433

22 Quadratic Applications 459

Part VI Finite Fields

23 Congruence Classes Modulo a Polynomial 479

24 Homomorphisms and Finite Fields 495

25 BCH Codes 511

Part VII Factoring Polynomials

26 Factoring in Z[x] 531

27 Irreducible Polynomials 557

Answers and Hints to the Exercises 569

References 595

Index 599

Editorial 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

From the reviews of the third edition:

“This book is an introduction to abstract algebra. … it has enough material to give the instructor flexibility in the course. It is a good book for the serious student to have in his/her library. … I would recommend this especially for self-study, as the book reads exactly as a good teacher talks to a class.” (Philosophy, Religion and Science Book Reviews, bookinspections.wordpress.com, October, 2013)

"This book can serve as both an introduction to number theory and abstract algebra, sacrifices have to be made with respect to its algebraic content. … the book has been written with a high degree of rigor and accuracy and I definitely recommend it for consideration as the basis of an alternative route into abstract algebra and its applications." (The Mathematical Association of America, April, 2009)

"The target audience remains students requiring a substantial introduction to the elements of university-level algebra. … the text proceeds throughout on a foundation built from the students’ familiarity with integers and polynomials over fields. Great care is taken to proceed to abstract concepts by way of familiar examples, and agreat many exercises are provided throughout the text. … A noteworthy feature of the book is the inclusion of extensive material on applications, to such topics as cryptography and factoring polynomials." (Kenneth A. Brown, Mathematical Reviews, Issue 2009 i)

From the Publisher