Algebra and Number Theory : An Integrated Approach

by Martyn Dixon (Univ. of Alabama, Tuscaloosa ); Leonid Kurdachenko (Univ. of Dnepropetrovsk, Ukraine ); Igor Subbotin (National University )

• ISBN13: 9780470640531
• ISBN10: 0470640537
• Format: eBook
• Copyright: 2010-12-01
• Publisher: Wiley

Summary

Explore the main algebraic structures and number systems that play a central role across the field of mathematicsAlgebra and number theory are two powerful branches of modern mathematics at the forefront of current mathematical research, and each plays an increasingly significant role in different branches of mathematics, from geometry and topology to computing and communications. Based on the authors' extensive experience within the field, Algebra and Number Theory has an innovative approach that integrates three disciplines--linear algebra, abstract algebra, and number theory--into one comprehensive and fluid presentation, facilitating a deeper understanding of the topic and improving readers' retention of the main concepts.The book begins with an introduction to the elements of set theory. Next, the authors discuss matrices, determinants, and elements of field theory, including preliminary information related to integers and complex numbers. Subsequent chapters explore key ideas relating to linear algebra such as vector spaces, linear mapping, and bilinear forms. The book explores the development of the main ideas of algebraic structures and concludes with applications of algebraic ideas to number theory.Interesting applications are provided throughout to demonstrate the relevance of the discussed concepts. In addition, chapter exercises allow readers to test their comprehension of the presented material.Algebra and Number Theory is an excellent book for courses on linear algebra, abstract algebra, and number theory at the upper-undergraduate level. It is also a valuable reference for researchers working in different fields of mathematics, computer science, and engineering as well as for individuals preparing for a career in mathematics education.

Table of Contents

Preface

Sets

Operations on Sets

Set mappings

Products of Mappings

Some properties of integers

Matrices and Determinants

Operations on matrices

Permutations of finite sets

Determinants of matrices

Computing Determinants

Properties of the product of matrices

Fields

Binary Algebraic Operations

Basic Properties of Fields

The Field of Complex Numbers

Vector Spaces

Vector Spaces

Dimension

The Rank of a Matrix

Quotient Spaces

Linear Mappings

Linear Mappings

Matrices of Linear Mappings

Systems of Linear Equations

Eigenvectors and eigenvalues

Bilinear Forms

Bilinear Forms

Classical Forms

Symmetric forms over R

Euclidean Spaces

Rings

Rings, Subrings and Examples

Equivalence Relations

Ideals and Quotient Rings

Homomorphisms of rings

Rings of polynomials and formal power series

Rings of multivariable polynomials

Groups

Groups and Subgroups

Examples of Groups and Subgroups

Cosets

Normal subgroups and Factor groups

Homomorphisms of Groups

Arithmetic Properties of Rings

Extending Arithmetic to Commutative Rings

Euclidean Rings

Irreducible Polynomials

Arithmetic Functions

Congruences

The Real Number System

The Natural Numbers

The Integers

The Rationals

The Real Numbers

Answers to selected exercises

Index

Table of Contents provided by Publisher. All Rights Reserved.