A Concrete Introduction to Higher Algebra, 2nd Edition

by Lindsay N. Childs

This book is written as an introduction to higher algebra for students with a background of a year of calculus. The first edition of this book emerged from a set of notes written in the 1970sfor a sophomore-junior level course at the University at Albany entitled "Classical Algebra." The objective of the course, and the book, is to give students enough experience in the algebraic theory of the integers and polynomials to appre­ ciate the basic concepts of abstract algebra. The main theoretical thread is to develop algebraic properties of the ring of unique factorization into primes, congruences and congruence classes, Fermat's theorem, the Chinese remainder theorem; and then again for the ring of polynomials. Doing so leads to the study of simple field extensions, and, in particular, to an exposition of finite fields. Elementary properties of rings, fields, groups, and homomorphisms of these objects are introduced and used as needed in the development. 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 elemen­ tary and computational number theory. A student who asks, "Why am I learning this?," willfind answers usually within a chapter or two. For a first course in algebra, the book offers a couple of advantages. • By building the algebra out of numbers and polynomials, the book takes maximal advantage of the student's prior experience in algebra and arithmetic. New concepts arise in a familiar context.

Genres: Mathematics, Textbooks, Algebra, Reference

522 pages, Paperback

First published December 1, 1978

Reviews

[Dave Neary · Rating 4 out of 5]

I really like this book! It switches from more elementary number theory (modular arithmetic, fundamental theorem of arithmetic) and more abstract algebra topics (groups, rings, fields). It has a light coverage of some applications like public key cryptography and primality tests, but does not give all the prerequisites for higher topics like elliptic curves and modular forms, which also depend on some complicated analysis too.