A Concrete Introduction to Higher Algebra

Lindsay N. Childs (eds.)0387745270, 978-0-387-74527-5, 978-0-387-74725-5

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.

“At every stage, a wide variety of applications is presented…The user-friendly exposition is appropriate for the intended audience”

– T.W. Hungerford, Mathematical Reviews

“The style is leisurely and informal, a guided tour through the foothills, the guide unable to resist numerous side paths and return visits to favorite spots…”

– Michael Rosen, American Mathematical Monthly

Table of contents :
Front Matter….Pages i-xiv
Numbers….Pages 3-7
Induction….Pages 9-25
Euclid’s Algorithm….Pages 27-52
Unique Factorization….Pages 53-70
Congruence….Pages 71-89
Congruence Classes….Pages 93-121
Rings and Fields….Pages 123-146
Matrices and Codes….Pages 147-167
Fermat’s and Euler’s Theorems….Pages 171-200
Applications of Euler’s Theorem….Pages 201-221
Groups….Pages 223-252
The Chinese Remainder Theorem….Pages 253-281
Polynomials….Pages 285-293
Unique Factorization….Pages 295-306
The Fundamental Theorem of Algebra….Pages 307-338
Polynomials in ℚ[ x ]….Pages 339-353
Congruences and the Chinese Remainder Theorem….Pages 355-371
Fast Polynomial Multiplication….Pages 373-383
Cyclic Groups and Cryptography….Pages 387-412
Carmichael Numbers….Pages 413-431
Quadratic Reciprocity….Pages 433-457
Quadratic Applications….Pages 459-475
Congruence Classes Modulo a Polynomial….Pages 479-493
Homomorphisms and Finite Fields….Pages 495-510
BCH Codes….Pages 511-527
Factoring in ℤ[ x ]….Pages 531-556
Irreducible Polynomials….Pages 557-567
Back Matter….Pages 569-603