A Course on Abstract Algebra / Minking Eie, & Shou-Te Chang

TABLE OF CONTENTS Machine generated contents note: 1.Preliminaries
1.1.Basic Ideas of Set Theory
1.2.Functions
1.3.Equivalence Relations and Partitions
1.4.A Note on Natural Numbers
Review Exercises
2.Algebraic Structure of Numbers
2.1.The Set of Integers
2.2.Congruences of Integers
2.3.Rational Numbers
Review Exercises
3.Basic Notions of Groups
3.1.Definitions and Examples
3.2.Basic Properties
3.3.Subgroups
3.4.Generating Sets
Review Exercises
4.Cyclic Groups
4.1.Cyclic Groups
4.2.Subgroups of Cyclic Groups
Review Exercises
5.Permutation Groups
5.1.Symmetric Groups
5.2.Dihedral Groups
5.3.Alternating Groups
Review Exercises
6.Counting Theorems
6.1.Lagrange's Theorem
6.2.Conjugacy Classes of a Group
Review Exercises
7.Group Homomorphisms
7.1.Examples and Basic Properties
7.2.Isomorphisms
7.3.Cayley's Theorem
Review Exercises
8.The Quotient Group
8.1.Normal Subgroups
8.2.Quotient Groups Note continued: 8.3.Fundamental Theorem of Group Homomorphisms
Review Exercises
9.Finite Abelian Groups
9.1.Direct Products of Groups
9.2.Cauchy's Theorem
9.3.Structure Theorem of Finite Abelian Groups
Review Exercises
10.Group Actions
10.1.Definition and Basic Properties
10.2.Orbits and Stabilizers
10.3.Burnside's Formula
Review Exercises
11.Sylow Theorems and Applications
11.1.The Three Sylow Theorems
11.2.Applications of Sylow Theorems
Review Exercises
12.Introduction to Group Presentations
12.1.Free Groups and Free Abelian Groups
12.2.Generators and Relations
12.3.Classification of Finite Groups of Small Orders
Review Exercises
13.Types of Rings
13.1.Definitions and Examples
13.2.Matrix Rings
Review Exercises
14.Ideals and Quotient Rings
14.1.Ideals
14.2.Quotient Rings
Review Exercises
15.Ring Homomorphisms
15.1.Ring Homomorphisms
15.2.Direct Products of Rings Note continued: 15.3.The Quotient Field of an Integral Domain
Review Exercises
16.Polynomial Rings
16.1.Polynomial Rings in the Indeterminates
16.2.Properties of the Polynomial Rings of One Variable
16.3.Principal Ideal Domains and Euclidean Domains
Review Exercises
17.Factorization
17.1.Irreducible and Prime Elements
17.2.Unique Factorization Domains
17.3.Polynomial Extensions of Factorial Domains
Review Exercises
18.Introduction to Modules
18.1.Modules and Submodules
18.2.Linear Maps and Quotient Modules
18.3.Direct Sums of Modules
Review Exercises
19.Free Modules
19.1.Free Modules
19.2.Determinant
Review Exercises
20.Vector Spaces over Arbitrary Fields
20.1.A Brief Review on Vector Spaces
20.2.A Brief Review on Linear Transformations
Review Exercises
21.Field Extensions
21.1.Algebraic or Transcendental?
21.2.Finite and Algebraic Extensions Note continued: 21.3.Construction with Straightedge and Compass
Review Exercises
22.All About Roots
22.1.Zeros of Polynomials
22.2.Uniqueness of Splitting Fields
22.3.Algebraically Closed Fields
22.4.Multiplicity of Roots
22.5.Finite Fields
Review Exercises
23.Galois Pairing
23.1.Galois Groups
23.2.The Fixed Subfields of a Galois Group
23.3.Fundamental Theorem of Galois Pairing
Review Exercises
24.Applications of the Galois Pairing
24.1.Fields of Invariants
24.2.Solvable Groups
24.3.Insolvability of the Quintic
Review Exercises