Textbook

# Algebra and Number Theory: An Integrated Approach

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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PREFACE.

CHAPTER 1 SETS.

1.1 Operations on Sets.

Exercise Set 1.1.

1.2 Set Mappings.

Exercise Set 1.2.

1.3 Products of Mappings.

Exercise Set 1.3.

1.4 Some Properties of Integers.

Exercise Set 1.4.

CHAPTER 2 MATRICES AND DETERMINANTS.

2.1 Operations on Matrices.

Exercise Set 2.1.

2.2 Permutations of Finite Sets.

Exercise Set 2.2.

2.3 Determinants of Matrices.

Exercise Set 2.3.

2.4 Computing Determinants.

Exercise Set 2.4.

2.5 Properties of the Product of Matrices.

Exercise Set 2.5.

CHAPTER 3 FIELDS.

3.1 Binary Algebraic Operations.

Exercise Set 3.1.

3.2 Basic Properties of Fields.

Exercise Set 3.2.

3.3 The Field of Complex Numbers.

Exercise Set 3.3.

CHAPTER 4 VECTOR SPACES.

4.1 Vector Spaces.

Exercise Set 4.1.

4.2 Dimension.

Exercise Set 4.2.

4.3 The Rank of a Matrix.

Exercise Set 4.3.

4.4 Quotient Spaces.

Exercise Set 4.4.

CHAPTER 5 LINEAR MAPPINGS.

5.1 Linear Mappings.

Exercise Set 5.1.

5.2 Matrices of Linear Mappings.

Exercise Set 5.2.

5.3 Systems of Linear Equations.

Exercise Set 5.3.

5.4 Eigenvectors and Eigenvalues.

Exercise Set 5.4.

CHAPTER 6 BILINEAR FORMS.

6.1 Bilinear Forms.

Exercise Set 6.1.

6.2 Classical Forms.

Exercise Set 6.2.

6.3 Symmetric Forms over R.

Exercise Set 6.3.

6.4 Euclidean Spaces.

Exercise Set 6.4.

CHAPTER 7 RINGS.

7.1 Rings, Subrings, and Examples.

Exercise Set 7.1.

7.2 Equivalence Relations.

Exercise Set 7.2.

7.3 Ideals and Quotient Rings.

Exercise Set 7.3.

7.4 Homomorphisms of Rings.

Exercise Set 7.4.

7.5 Rings of Polynomials and Formal Power Series.

Exercise Set 7.5.

7.6 Rings of Multivariable Polynomials.

Exercise Set 7.6.

CHAPTER 8 GROUPS.

8.1 Groups and Subgroups.

Exercise Set 8.1.

8.2 Examples of Groups and Subgroups.

Exercise Set 8.2.

8.3 Cosets.

Exercise Set 8.3.

8.4 Normal Subgroups and Factor Groups.

Exercise Set 8.4.

8.5 Homomorphisms of Groups.

Exercise Set 8.5.

CHAPTER 9 ARITHMETIC PROPERTIES OF RINGS.

9.1 Extending Arithmetic to Commutative Rings.

Exercise Set 9.1.

9.2 Euclidean Rings.

Exercise Set 9.2.

9.3 Irreducible Polynomials.

Exercise Set 9.3.

9.4 Arithmetic Functions.

Exercise Set 9.4.

9.5 Congruences.

Exercise Set 9.5.

CHAPTER 10 THE REAL NUMBER SYSTEM.

10.1 The Natural Numbers.

10.2 The Integers.

10.3 The Rationals.

10.4 The Real Numbers.

INDEX.

Author Information
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.

Reviews

“The book is well-written and covers, with plenty of exercises, the material needed in the three aforementioned courses, albeit in a new order.”  (Zentralblatt MATH, 1 December 2012)

"However, instructors contemplating such a unified approach should give this book serious consideration. Recommended. Upper-division undergraduates through researchers/faulty." (Choice , 1 April 2011)

## Available Versions

Algebra and Number Theory: An Integrated Approach
by Martyn Dixon, Leonid Kurdachenko, Igor Subbotin
ISBN 978-0-470-49636-7
September 2010
Hardcover, 523 pages