A Concrete Introduction to Higher Algebra

✍ Scribed by Lindsay N. Childs (auth.)

• Publisher: Springer-Verlag New York
• Year: 1995
• Tongue: English
• Leaves: 539
• Series: Undergraduate Texts in Mathematics
• Edition: 2
• Category: Library

✦ Synopsis

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 integers: 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.

✦ Table of Contents

Front Matter....Pages i-xv
Numbers....Pages 1-7
Induction....Pages 8-24
Euclid’s Algorithm....Pages 25-46
Unique Factorization....Pages 47-62
Congruences....Pages 63-75
Congruence Classes....Pages 76-90
Applications of Congruences....Pages 91-117
Rings and Fields....Pages 118-133
Fermat’s and Euler’s Theorems....Pages 134-154
Applications of Fermat’s and Euler’s Theorems....Pages 155-179
On Groups....Pages 180-193
The Chinese Remainder Theorem....Pages 194-207
Matrices and Codes....Pages 208-230
Polynomials....Pages 231-238
Unique Factorization....Pages 239-252
The Fundamental Theorem of Algebra....Pages 253-276
Derivatives....Pages 277-285
Factoring in ℚ[ x ], I....Pages 286-292
The Binomial Theorem in Characteristic p ....Pages 293-301
Congruences and the Chinese Remainder Theorem....Pages 302-309
Applications of the Chinese Remainder Theorem....Pages 310-322
Factoring in F p [ x ] and in ℤ[ x ]....Pages 323-345
Primitive Roots....Pages 346-352
Cyclic Groups and Primitive Roots....Pages 353-362
Pseudoprimes....Pages 363-377
Roots of Unity in ℤ/ m ℤ....Pages 378-396
Quadratic Residues....Pages 397-413
Congruence Classes Modulo a Polynomial....Pages 414-431
Some Applications of Finite Fields....Pages 432-463
Classifying Finite Fields....Pages 464-482
Back Matter....Pages 483-524

✦ Subjects

Algebra