Course overview
This subject presents the foundational material for the last of the basic algebraic structures pervading contemporary pure mathematics, namely fields and modules. The basic definitions and elementary results are given, followed by two important applications of the theory: to the classification of finitely generated abelian groups, and to Jordan canonical form for matrices. The subject concludes by returning to fields to present interesting applications of the theory.
Fields: vector spaces, matrices, characteristic values; extension fields. Modules: finitely generated modules over a PID; canonical forms for matrices; Jordan canonical form. Applications of fields to algebraic and geometric problems.
Course learning outcomes
- Demonstrate understanding of the concepts of a field and a module and their role in mathematics.
- Demonstrate familiarity with a range of examples of these structures.
- Prove the basic results of field theory and module theory.
- Explain the structure theorem for finitely generated modules over a principal ring and its applications to abelian groups and matrices.
- Apply the theory in the course to solve a variety of problems at an appropriate level of difficulty.
- Demonstrate skills in communicating mathematics orally and in writing.