A Concrete Introduction to Higher Algebra

Bol

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. Over 900 exercises are found throughout the book. 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 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. The new examples and theory are built in a well-motivated fashion and made relevant by many applications - to cryptography, coding, integration, history of mathematics, and especially to elementary and computational number theory. The later chapters include expositions of Rabiin's probabilistic primality test, quadratic reciprocity, and the classification of finite fields. Over 900 exercises are found throughout the book.

Bol Partner

This is an introduction to higher algebra for students with a background of a year of calculus. The book aims to give these students enough experience in the algebraic theory of the integers and polynomials to appreciate 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 congrunce classes; Fermat's theorem; the Chinese remainder theorem; and for the ring of polynomials. 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 elementary and computational number theory. Many of the recent advances in computational number theory are built on the mathematics which is presented in this book. Thus the book may be used as a first course in higher algebra, as originally intended, but may also serve as an introduction to modern computational number theory, or to applied algebra.