What are you going to learn?
- 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
Content
Chapter 1. Introduction
- Logic, Sets, Operations
- Equivalence Relations
- Equivalence Classes
- Permutations Groups
- Subgroups
- Groups and Symmetry
Chapter 2. Groups 1
- Definition. Examples
- Permutations
- Subgroups
- Groups and Symmetry
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