## Math 4033 Advanced Algebra II

* Three credits. Three hours of lecture per week. Prerequisite: MATH 4032*

Introduction to rings. Subrings. Ring homomorphisms. Ideals and quotient rings. Polynomial rings. The field of fractions of an integral domain. Introduction to field theory.

- Preliminaries
- Review of Group Theory.
- Subgroups.
- Cyclic groups.
- Cosets.
- Normal subgroups.
- Factor groups.
- Homomorphism and isomporphism theorems for groups.
- Correspondence theorem for groups.
- Cauchy’s theorem for abelian groups.

- Introduction to Ring Theory
- Definition of a ring.
- Some examples of rings.
- Division rings, fields.
- Skew fields. Quarternions.
- Integral domains.
- A finite integral domain is a field.
- The integers modulo p where p is a prime.

- Subrings
- A subring criterion.
- Homomorphisms
- Ideals.
- Maximal and prime ideals.
- Quotient rings.
- Homomorphism and isomorphism theorems for rings.
- Correspondence theorem for rings.
- The Gaussian Integers.
- Field homomorphisms.
- The group of fractions of an abelian monoid with cancellation.
- The field of fractions.

- Polynomials
- Polynomial rings.
- The division theorem.

- Euclidean domains.
- Examples of Euclidean domains.
- The integers (with d(a) = |a|).
- The Gaussian integers.
- F[X] where F is a field.

- Euclids algorithm for Euclidean domains.

- Examples of Euclidean domains.
- Principal ideal domain.
- Divisibility, greatest common factors
- Least common multiples.

- Unique factorization domains
- Irreducibility criteria
- ED ==> PID ==> UFD

- Fields
- Prime fields sub fields
- Simple field extensions
- Splitting fields
- Algebraic Closurev

### Text

I. Herstein, *Abstract Algebra*, MacMillan Publishing Company, 1990.

### Bibliography

- I. Herstein,
*Topics in Algebra*, Wiley. - J.B. Fraleigh with Historical Notes by Victor Katz.
*A First Course in Abstract Algebra*, (5th edition), Addison-Wesley, 1994. - D. Maclane and G. Birkhoff,
*Algebra*. 3rd Ed., Chelsea Pubishing Company, New York, N. Y., 1988.