What are you going to learn?
Content
Define a group, give examples of groups, list properties that hold in every group and state definitions of particular features of groups
Define a ring, give examples of rings, list properties that hold in every ring and state definitions of particular features of rings
Define a field, give examples of fields, list properties that hold in every field and state definitions of particular features of fields
Prove statements about these mathematical structures
Chapter 1. Introduction
Logic, Sets, Operations
Equivalence Relations
Equivalence Classes
Permutations Groups
Subgroups
Groups and Symmetry
Chapter 2. Groups 1
Chapter 3. Groups 2
Properties
Generators. Direct Groups
Cosets
Lagranges's Theorem. Cyclic Groups
Isomorphisms
Cayle's Theorem
Chapter 4. Groups Homomorphisms
Homomorphisms and Kernels
Quotient Groups
The Fundamental Homomorphism Theorem
Chapter 5. Introduction to Rings
Definition and Examples
Integral Domains, Division Rings
Fields
Isomorphisms. Characteristics
Chapter 6. Numbers Systems
Ordered Integral Domains
Integers
Field of Quotients. Field of Rationals
Ordered Fields. The Field of Reals
The Field of Complex Numbers
Complex Roots of Unity
Chapter 7. Polynomials
Definitions and Elementary properties
The Division Algorithm
Factorization of Polynomials
Unique Factorization Domains
Chapter 8. Quotient Rings
Homomorphisms of Rings. Ideals
Quotient Rings
Factorization and Ideals
Chapter 9. Galois Theory
Simple Extensions. Degree
Roots of Polynomials
Fundamental Theorem
Algebraic Extensions
Splitting Fields. Galois Groups
Separability and Normality
Fundamental Theorem of Galois Theory
Solvability by Radicals
Finite Fields
Bibliography
Durbin, J.R. A Modern Algebra:An Introduction. 6th Ed. John Wiley and Sons, 2009.
Rotman, J.J. A First Course in Abstract Algebra. 7th Ed. Prentice Hall
