Algebra and Number Theory: An Integrated Approach

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John Wiley & Sons, Jul 15, 2011 - Mathematics - 544 pages
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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
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About the author (2011)

MARTYN R. DIXON, PhD, is Professor in the Department of Mathematics at the University of Alabama, Tuscaloosa. He has authored more than sixty published journal articles on infinite group theory, formation theory and Fitting classes, wreath products, and automorphism groups.

LEONID A. KURDACHENKO, PhD, is Distinguished Professor and Chair of the Department of Algebra at the Dnepropetrovsk National University (Ukraine). Dr. Kurdachenko has authored more than 150 journal articles on the topics of infinite-dimensional linear groups, infinite groups, and module theory.

IGOR YA. SUBBOTIN, PhD, is Professor in the Department of Mathematics and Natural Sciences at National University (California). Dr. Subbotin is the author of more than 100 published journal articles on group theory, cybernetics, and mathematics education.

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