Pub. Date:
Springer New York
Concrete Introduction to Higher Algebra / Edition 3

Concrete Introduction to Higher Algebra / Edition 3

by Lindsay N. Childs


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Concrete Introduction to Higher Algebra / Edition 3

This book is an informal and readable introduction to higher algebra at the post-calculus level. The concepts of ring and field are introduced through study of the familiar examples of the integers and polynomials. A strong emphasis on congruence classes leads in a natural way to finite groups and finite fields. The new examples and theory are built in a well-motivated fashion and made relevant by many applications - to cryptography, error correction, integration, and especially to elementary and computational number theory. The later chapters include expositions of Rabin's probabilistic primality test, quadratic reciprocity, the classification of finite fields, and factoring polynomials over the integers. Over 1000 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. The new edition includes topics such as Luhn's formula, Karatsuba multiplication, quotient groups and homomorphisms, Blum-Blum-Shub pseudorandom numbers, root bounds for polynomials, Montgomery multiplication, and more.

Product Details

ISBN-13: 2901441925618
Publisher: Springer New York
Publication date: 11/19/2010
Series: Undergraduate Texts in Mathematics Series
Edition description: Softcover reprint of hardcover 3rd ed. 2009
Pages: 604
Product dimensions: 6.00(w) x 1.25(h) x 9.00(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

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