**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*.Rotman, J.J.

. 7th Ed. Prentice Hall*A First Course in Abstract Algebra*