Algebra Examination Syllabus
Topics
Group Theory
Groups and Homomorphisms. The Isomorphism Theorems. Permutation Groups. G-sets. Direct Products. The Sylow Theorems. The Fundamental Theorem of Finitely Generated Abelian Groups. The Jordan-Holder Theorem. Solvable Groups.
Rings and Modules
Principal Ideal Domains, Unique Factorization Domains. Rings of Quotients (Fractions). Modules, Endomorphism Rings of Modules. Direct Product and Direct Coproducts of Rings and Modules. Socles and Radicals of Rings and Modules. Artinian and Noetherian Rings and Modules, Wedderburn-Artin Structure Theorem. Free Modules, Injective and Projective Modules. Von Neumann Regular Rings.
Field Theory
Theory of Algebraic Extensions of fields. Normal and Separable Extensions. Finite Fields. Fundamental Theorem of Galois Theory and its Applications. Solvability by Radicals, Constructibility by straight edge and compass.
Linear Algebra
Vector Spaces, Basis. Dual Vector Spaces. Matrices and Linear Mappings. Bilinear Forms. Inner Products. Canonical Forms - Diagonal, Triangular, Jordan.
Bibliography
- Joseph J. Rotman, "The Theory of Groups: An Introduction", Allyn and Bacon, Inc. (Chapters 1-5, and appropriate sections in Chapter 6).
- N.H. McCoy, "The Theory of Rings", The McMillan Co. (Chapters 1,2 and appropriate sections in Chapters 3-5).
- P.B. Bhattacharya, S.K. Jain, and S.R. Nagpaul, "Basic Abstract Algebra", Cambridge University Press, second edition, 1994.
- S. Lang, "Linear Algebra", Addison-Wesley Publishing Company (Appropriate material in part one and part two).
- S. Lang, "Algebra", Addison-Wesley, Chapters 1-3, Chap. 5-7, appropriate sections in Chapter 8.
- E. Artin, "Galois Theory", Notre Dame Mathematical Lectures No. 2 (Complete book).
- N. Jacobson, "Basic Algebra 1", W.H. Freeman and Company (Appropriate material).
- P.M. Cohn, "Algebra I and II", John Wiley & Sons (Appropriate material).
- I.M. Isaacs, "Algebra, a Graduate Course"
- I. Kaplansky, "Fields and Rings", Chicago University Press, 1989.