Abstract Algebra

Title
Preface
Contents
0. Preliminaries
0.1. Set theoretic notations
0.2. Correspondences, maps, relations
0.3. Cartesian product and universal mapping properties
0.4. Cardinal numbers
0.5. Zorn's lemma, axiom of choice, well-ordering axiom
References
I. Natural Numbers, Integers, and Rational Numbers
I.1. Peano's axioms
I.2. Addition in P
I.3. Ordering in P: A second definition of finiteness
I.4. Multiplication in P
I.5. Construction of the integers Z
I.6. Divisibility theory in Z
I.7. Construction of the rational numbers Q
References
II. Groups, Rings, Integral Domains, Fields
II.1. Multiplicative systems: Groups
II.2. Homomorphisms
II.3. Rings, integral domains, and fields
II.4. Polynomial rings
References
III. Elementary Theory of Groups
III.1. Basic concepts
III.2. Homomorphisms of groups
Excursion I
III.3. Transformation groups: Sylow's theorem
III.4. The finite symmetric groups Sn
Excursion II
III.5. Direct product of groups: Fundamental Theorem of Finite Abelian Groups
Excursion III
References
IV. Elementary Theory of Rings
IV.1. Basic concepts
IV.2. Divisibility theory in integral domains
IV.3. Fields of rational functions: Partial fraction decomposition
IV.4. Modules and their endomorphism rings. Matrices
IV.5. Rings of functions
References
V. Modules and Associated Algebras over Commutative Rings
V.1. Tensor product of modules over commutative rings
V.2. Free modules over a PID
Excursion IV
V.3. Modules of finite type over a PID
Excursion V
V.4. Tensor algebras, exterior algebras, and determinants
V.5. Derivations, traces, and characteristic polynomials
Excursion VI
V.6. Dual modules
References
VI. Vector Spaces
VI.1. Basic concepts
VI.2. Systems of linear equations
VI.3. Decomposition of a vector space with respect to a linear endomorphism
VI.4. Canonical forms of matrices: Characteristic values: Characteristic vectors
Excursion VII
References
VII. Elementary Theory of Fields
VII.1. Basic concepts
VII.2. Algebraic extension fields: Splitting fields
VII.3. Algebraically closed fields: Algebraic closure
VII.4. Algebraic independence: Purely transcendental extensions: Transcendence base
Excursion VIII
VII.5. Separable and inseparable algebraic field extensions
VII.6. Finite fields: Primitive element theorem
References
VIII. Galois Theory
VIII.1. Basic concepts
VIII.2. Fundamental theorems
Excursion IX
Excursion X
VIII.3. Solvability of polynomial equations by radicals
VIII.4. Cyclotomic polynomials over Q: Kummer extensions
References
IX. Real and Complex Numbers
IX.1. Construction of real and complex numbers
IX.2. Fundamental Theorem of Algebra
Excursion XI
Excursion XII
References
Index
Symbols and notations