A Concrete Introduction To Higher Algebra

by Childs, Lindsay N.

• ISBN13: 9780387745275
• ISBN10: 0387745270
• Edition: 3rd
• Format: Hardcover
• Copyright: 2008-12-05
• Publisher: Springer Nature

Summary

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, 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, 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.

Table of Contents

Numbers.- Induction.- Euclid's Algorithm.- Unique Factorization.- Congruences.- Congruence Classes.- Applications of Congruences.- Rings and Fields.- Fermat's and Euler's Theorems.- Applications of Fermat's and Euler's Theorems.- On Groups.- The Chinese Remainder Theorem.- Matrices and Codes.- Polynomials.- Unique Factorization.- The Fundamental Theorem of Algebra.- Derivatives.- Factoring in Qx,I.- The Binomial Theorem in Characteristic p.- Congruences and the Chinese Remainder Theorem.- Applications of the Chinese Remainder Theorem.- Factoring in Fpx and in Zx.- Primitive Roots.- Cyclic Groups and Primitive Roots.- Pseudoprimes.- Roots of Unity in Z/mZ.- Quadratic Residues.- Congruence Classes Mopdulo a Polynomial.- Some Applications of Finite Fields.- Classifying Finite Fields.