MAS 5312 Introduction to Algebra 2

The text for the course is Dummit and Foote, but the topics should be essentially the same whichever text, if any, is used.

This exam is on Ring theory. The exam will test for:

  • Ability to write clear, rigorous proofs.
  • A solid understanding of the basic definitions.
  • Thorough understanding of the statements and proofs of theorems.
  • Ability to apply theorems to specific situations.
  • Familiarity with a broad range of examples and ability to do computations with them.

All topics are examinable except for those explicitly excluded. In the case of very long proofs, exam questions may involve only a part of the argument. In exams it is generally allowed to apply main theorems by quoting them correctly, as long as doing so does not result in circular logic, as would be the case if the question is really asking for part of the proof of the theorem quoted. It is possible that standard material from the first semester of the First Year Algebra sequence will be needed to solve some problems.

The modules below (in order) correspond to chapters 7–13 of Dummit and Foote.

  1. Topics: Rings, polynomial rings, matrix rings, subrings, homomorphisms, ideals, quotient rings, maximal ideals, prime ideals, rings of fractions, Chinese remainder Theorem.
  2. Topics: Euclidean domains, PIDs, UFDs.
  3. Topics: Polynomial rings, polynomial rings over fields, factorization and irreducibility.
  4. Section 10.4 on tensor products and section 10.5 on exact sequences, projective, injective and flat modules are excluded.
    Topics: Modules, submodules, module homomorphisms, quotient modules, generation of modules, direct sums, free modules (optional, not examinable).
  5. Topics from chapter 11 familiar to students may be covered in less detail. Section 11.4 on determinants is optional and section 11.5 on Tensor algebras is excluded.
    Topics: Vector spaces. Basic theory of vector spaces, matrix of a linear transformation, dual vector spaces.
  6. Topics: Fundamental theorem for finitely generated modules over a PID, rational canonical form, Jordan Canonical form.
  7. Chapter 13 is covered only up to 13.4.
    Topics: Field extensions, algebraic extensions, straight edge and compass constructions (optional), splitting fields (optional, not examinable), fundamental theorem of algebra (optional, not examinable).

Two sets of practice problems are available: