Math 4033 Advanced Algebra II

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.

  1. Preliminaries
    1. Review of Group Theory.
    2. Subgroups.
    3. Cyclic groups.
    4. Cosets.
    5. Normal subgroups.
    6. Factor groups.
    7. Homomorphism and isomporphism theorems for groups.
    8. Correspondence theorem for groups.
    9. Cauchy’s theorem for abelian groups.
  2. Introduction to Ring Theory
    1. Definition of a ring.
    2. Some examples of rings.
    3. Division rings, fields.
    4. Skew fields. Quarternions.
    5. Integral domains.
    6. A finite integral domain is a field.
    7. The integers modulo p where p is a prime.
  3. Subrings
    1. A subring criterion.
    2. Homomorphisms
    3. Ideals.
    4. Maximal and prime ideals.
    5. Quotient rings.
    6. Homomorphism and isomorphism theorems for rings.
    7. Correspondence theorem for rings.
    8. The Gaussian Integers.
    9. Field homomorphisms.
    10. The group of fractions of an abelian monoid with cancellation.
    11. The field of fractions.
  4. Polynomials
    1. Polynomial rings.
    2. The division theorem.
  5. Euclidean domains.
    1. Examples of Euclidean domains.
      • The integers (with d(a) = |a|).
      • The Gaussian integers.
      • F[X] where F is a field.
    2. Euclids algorithm for Euclidean domains.
  6. Principal ideal domain.
    1. Divisibility, greatest common factors
    2. Least common multiples.
  7. Unique factorization domains
    1. Irreducibility criteria
    2. ED ==> PID ==> UFD
  8. Fields
    1. Prime fields sub fields
    2. Simple field extensions
    3. Splitting fields
    4. Algebraic Closurev

Text

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

Bibliography

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