Algebra and Number Theory – An Integrated Approach

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.

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. ANSWERS TO SELECTED EXERCISES. INDEX.