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A Concrete Introduction to Higher Algebra

A Concrete Introduction to Higher Algebra
Catalogue Information
Field name Details
Dewey Class 512
Title A Concrete Introduction to Higher Algebra (EB) / edited by Lindsay N. Childs.
Author Childs, Lindsay N.
Other name(s) SpringerLink (Online service)
Publication New York, NY : : Springer New York, , 2009.
Physical Details XIV, 603 p. : online resource.
Series Undergraduate texts in mathematics 0172-6056
ISBN 9780387747255
Summary Note 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:
Contents note Preface -- Numbers -- Induction -- Euclid's Algorithm -- Unique Factorization -- Congruence -- Congruence Classes -- Rings and Fields -- Matrices and Codes -- Fermat's and Euler's Theorems -- Applications of Fermat's and Euler's Theorems -- Groups -- The Chinese Remainder Theorem -- Polynomials -- Unique Factorization -- The Fundamental Theorem of Algebra -- Polynomials in Q[x] -- Congruences and the CRT -- Fast Polynomial Multiplication -- Cyclic Groups and Cryptography -- Carmichael Numbers. -- Quadratic Reciprocity -- Quadratic Applications -- Congruence Classes Modulo a Polynomial -- Homomorphism and Finite Fields -- BCH Codes -- Factoring in Z[x] -- Irreducible Polynomials -- Answers and Hints to the Exercises -- References -- Index.-.
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