Introduction to Algebra (Universitext)

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