# Algebra and Number Theory: An Integrated Approach

John Wiley & Sons, Jul 15, 2011 - Mathematics - 544 pages
Explore the main algebraic structures and number systems that play a central role across the field of mathematics

Algebra 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.

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### Contents

 An Integrated Approach CHAPTER 2 MATRICES AND DETERMINANTS 41 An Integrated Approach CHAPTER 3 FIELDS 105 An Integrated Approach CHAPTER 4 VECTOR SPACES 145 An Integrated Approach CHAPTER 5 LINEAR MAPPINGS 187 An Integrated Approach CHAPTER 6 BILINEAR FORMS 226 An Integrated Approach CHAPTER 7 RINGS 272
 An Integrated Approach CHAPTER 8 GROUPS 338 An Integrated Approach CHAPTER 9 ARITHMETIC PROPERTIES OF RINGS 384 An Integrated Approach CHAPTER 10 THE REAL NUMBER SYSTEM 448 An Integrated Approach ANSWERS TO SELECTED EXERCISES 489 An Integrated Approach INDEX 513 Copyright