M3P8: Algebra III (Rings and Modules)

Fri 5-6 in Huxley 140, Mon 12-2 in Huxley 340
Dr. David Helm
672 Huxley
Office Hours: Fri 2-4 (or by appointment)

Course description

This course is an introduction to ring theory. The topics covered will include ideals, factorization, the theory of field extensions, finite fields, polynomial rings in several variables, and the theory of modules.

Suggested References

In addition to the lecture notes, chapters 10-13 of Algebra, First Edition, by Michael Artin (or chapters 11, 12, 14, and 15 of the second edition) cover much of the material we will be studying.
The course builds on the ring theory developed last year in weeks 9,10, and 11 of M2P2. Martin Liebeck has kindly made his notes on the subject available:
Week 9
Week 10
Week 11

Lecture Notes: Core Topics

These are last year's lecture notes for the core topics we will cover. These will cover about the first six or seven weeks of the course. There are still probably many typos; please let me know if you find any!
Part 1: Basic Definitions and Examples
Part 2: Homomorphisms, Ideals, and Quotients
Part 3: Factorization
Part 4: The Chinese Remainder Theorem
Part 5: Fields and Field Extensions
Part 6: Finite Fields
Part 7: R-Modules
Part 8: Noetherian Rings and Modules
Part 9: Polynomial Rings in Several Variables

Supplemental topics

In addition to the core topics, we will cover a few additonal topics drawn from the following list: Integral Extensions and Algebraic Integers, Modules over Principal Ideal Domains, Algebraic Geometry, and Noncommutative Ring Theory. Notes for these topics will be posted towards the end of the term.
Part 7a: Modules over a Euclidean domain
Part 10: Noncommutative Rings
Part 11: Simple Algebras and the Artin-Wedderburn Theorem

Assessed Coursework

There will be two assessed courseworks assigned, each worth 5 percent of your total marks. They should be handed in at the Undergraduate Office by 4pm on the day they are due. The first will be due on Friday, 1 November, and the second will be due on Friday, 29 November. On each due date the day's class will be a problem session dedicated to discussing the assigned questions (and possibly additional questions if time allows). The courseworks will be posted here at least two weeks in advance of the due dates.
Assessed Problems 1 (Due Friday 1 November, worth 5%) Solutions
Assessed Problems 2 (Due Friday 29 November, worth 5%)

Discussion Problems

In addition to the assessed coursework, I will be assigning additional coursework in the form of discussion problems. These are more challenging problems that will be discussed in problem sessions on the date they are due; they will not be assessed work. Since these questions are more difficult than typical assessment or exam questions, you are encouraged to work on them in groups and share ideas and approaches (and of course I am happy to give hints in office hours!) The discussion problems will be due on 25 October, 15 November, and 6 December and posted here at least two weeks before those dates.
Discussion Problems 1 (Due Friday 25 October, NOT ASSESSED) Solutions
Discussion Problems 2 (Due Friday 15 November, NOT ASSESSED) Solutions
Discussion Problems 3 (Due Friday 6 December, NOT ASSESSED) Solutions

Mastery Material

The mastery material is on algebraic integers and factorization, and is summarized in these lecture notes: Algebraic Integers and Factorization. An assortment of (entirely optional) exercises is here.