Algebra and Number Theory An Integrated Approach 1st Edition by Martyn Dixon, Leonid Kurdachenko, Igor Subbotin 0470496363 9780470496367
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Product details:
ISBN 10: 0470496363
ISBN 13: 9780470496367
Author: Martyn Dixon, Leonid Kurdachenko, Igor Subbotin
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.
Table of contents:
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
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Tags: Martyn Dixon, Leonid Kurdachenko, Igor Subbotin, Algebra