MA5301: Finite Groups and Finite Fields


  • Text I. Martin Isaacs , Algebra A Graduate Course, Brooks \& Cole, Pacific Grove California 1994.
  • Course Description. MA 5301 - Finite Groups and Finite Fields Basic theory of finite groups (subgroups, normality, homomorphisms, abelian groups, cyclic groups, commutators, order, cosets, index, conjugacy, simple groups, Sylow Theorems), basic theory of finite fields (prime fields, irreducible polynomials, Galois groups, trace), families of groups defined over finite fields (linear groups).
  • Syllabus
    We will discuss at least the following topics :
    1. Chapter 1: Definitions and Examples of Groups
    2. Chapter 2: Subgroups and Cosets
    3. Chapter 3: Homomorphisms
    4. Chapter 4: Group Actions
    5. Chapter 5: The Sylow Theorems and p-Groups
    6. Chapter 6: Permutation Groups
    7. Chapter 12: Rings, Ideals and Modules
    8. Chapter 16: Polynomial Rings, PIDs, and UFDs
    9. Chapter 17: Field Extensions
    10. Chapter 18: Galois Theory
    11. Chapter 20: Cyclotomy and Geometeric Constructions
    12. Chapter 21: Finite Fields
    13. Grading
      Your grade will be based on 3 take home examinations. a final examination.
    14. Start every written assignment at the top of a new page, put your name at the beginning of each page and do not staple different problems together. I expect the problems to be well written in full English sentences with no gaps in detail or logic. Please be as elegant and as concise as possible. Cite all references.