This textbook, written by a dedicated and successful pedagogue who developed the present undergraduate algebra course at Moscow State University, differs in several respects from other algebra textbooks available in English. The book reflects the Soviet approach to teaching mathematics with its emphasis on applications and problem-solving -- note that the mathematics department in Moscow is called the I~echanics-Mathematics" Faculty. In the first place, Kostrikin's textbook motivates many of the algebraic concepts by practical examples, for instance, the heated plate problem used to introduce linear equations in Chapter 1. In the second place, there are a large number of exercises, so that the student can convert a vague passive understanding to active mastery of the new ideas. Thes~ problems are intended to be challenging but doable by the student; the harder ones have hints at the back of the book. This feature also makes the book ideally suited for learning algebra on one's own outside of the framework of an organized course. In the third place, the author treats material which is usually not part of an elementary course but which is fundamental in applications. Thus, Part II includes an introduction to the classical groups and to representation theory. With many American colleges now trying to bring their undergraduate mathematics curriculum closer to applications, it seems worthwhile to translate Soviet textbooks which reflect their greater experience in this area of mathematical pedagogy.
Author(s): Aleksei Ivanovich Kostrikin
Edition: Softcover reprint of the original 1st ed. 1982
Publisher: Springer
Year: 1982
Language: English
Commentary: Originates from the same scan as https://libgen.is/book/index.php?md5=E39E53C26329FCE8ADF4FD64133133DC , but binarized and cleaned
Pages: 590
City: New York
TItle
A Note on the English Edition
Translator's Preface
Contents
Foreword
Advice to the Reader
Part One: Foundations of Algebra
Further Reading
Chapter 1. Sources of Algebra
§1. Algebra in brief
§2. Some model problems
§3. Systems of linear equations. The first steps
§4. Determinants of small order
§5. Sets and mappings
§6. Equivalence relations. Quotient maps
§7. The principle of mathematical induction
§8. Integer arithmetic
Chapter 2. Vector Spaces. Matrices
§1. Vector spaces
§2. The rank of a matrix
§3. Linear maps. Matrix operations
§4. The space of solutions
Chapter 3. Determinants
§1. Determinants: construction and basic properties
§2. Further properties of determinants
§3. Applications of determinants
Chapter 4. Algebraic Structures -- Groups, Rings, Fields
§1. Sets with algebraic operations
§2. Groups
§3. Morphisms of groups
§4. Rings and fields
Chapter 5. Complex Numbers and Polynomials
§1. The field of complex numbers
§2. Rings of polynomials
§3. Factoring in polynomial rings
§4. The field of fractions
Chapter 6. Roots of Polynomials
§1. General properties of roots
§2. Symmetric polynomials
§3. C is algebraically closed
§4. Polynomials with real coefficients
Part Two: Groups, Rings, Modules
Further Reading
Chapter 7. Groups
§1. Classical groups in low dimensions
§2. Group actions on sets
§3. Some group theoretic constructions
§4. The Sylow theorems
§5. Finite abelian groups
Chapter 8. Elements of Representation Theory
§1. Definitions and examples of linear representations
§2. Unitary and reducible representations
§3. Finite rotation groups
§4. Characters of linear representations
§5. Irreducihle representations of finite groups
§6. Representations of SU(2) and SO(3)
§7. Tensor products of representations
Chapter 9. Toward a Theory of Fields, Rings and Modules
§1. Finite field extensions
§2. Various results about rings
§3. Modules
§4. Algebras over a field
Appendix. The Jordan Normal Form of a Matrix
Hints to the Exercises
Index
