Math 533: Abstract Algebra I, Winter 2021
Professor: Darij Grinberg
Organization
Course description
An introduction to rings and modules, including the structure of finitely generated modules over a PID, tensor products and ideals. Polynomial rings, symmetric and exterior algebras and Gröbner bases will also be discussed, as will Galois theory if time allows.
Level: graduate.
Prerequisites: at least one semester of (undergraduate) abstract algebra.
Course materials
- Required:
- David S. Dummit, Richard M. Foote, Abstract algebra, 3rd edition, Wiley 2004: The book we shall use. Available at the usual places in various formats (e.g., PDF). Mind the errata. We will use Chapters 7-14.
- Other:
-
The following list gravitates towards freely available and new sources. See here or here or here or here for more standard references.
- Keith Conrad, Expository papers, 2020. Specifically the ones on "Ring Theory", "Linear/Multilinear algebra", "Fields and Galois theory". Conrad mostly sticks to the commutative case of everything, but what he explains he explains really well.
- Drew Armstrong, various classes, specifically Math 561 Fall 2018 (groups) and Math 562 Spring 2019 (rings). Like everything by Armstrong, these emphasize the geometry and the history. Highly recommended.
- Richard Elman, Lectures on Abstract Algebra, preliminary version 2020. Long set of notes; goes rather deep.
- Frederick M. Goodman, Algebra: Abstract and Concrete, edition 2.6, 2016. Relatively introductory, with a focus on groups, geometric symmetry and Galois theory.
- Mark Steinberger, Algebra, 2006.
- Anthony Knapp, Basic Algebra and Advanced Algebra, digital 2nd editions, 2016.
- Timothy J. Ford, Introduction to Abstract Algebra, 2020.
- Timothy J. Ford, Abstract Algebra, 2020.
- Emil Artin, Arthur N. Milgram, Galois Theory: Lectures Delivered at the University of Notre Dame, 1971. A classic introduction to Galois theory, freely available.
- Antoine Chambert-Loir, (Mostly) Commutative Algebra, 2020. Draft of a Bourbaki-style monograph (mostly above the level of this course).
- Alistair Savage, MAT 3143: Rings and Modules, 2020. Introductory notes.
- Peter J. Cameron, Introduction to Algebra, 2nd edition 2008. Well-regarded British text.
- James S. Milne, Fields and Galois Theory, 2020. Rather concise text written by a famous arithmetic geometer.
- David A. Cox, John Little, Donal O'Shea, Ideals, Varieties, and Algorithms, 4th edition 2015. An introduction to commutative algebra and algebraic geometry based on Gröbner bases. It starts rather elementary (with univariate polynomials).
- Alberto Elduque, Groups and Galois theory, 2019 and Alberto Elduque, Introduction to Algebra, 2017.
- Richard Koch, Galois Theory, 2017.
- Willem A. de Graaf, Lecture notes on algebra, 2000.
- Andrew Baker, An introduction to Galois theory, 2020 with solutions.
- Miles Reid, MA3D5 Galois theory, 2014.
Course calendar
- Note:
- The notes below have evolved into my Introduction to the algebra of rings and fields text (which includes all the material covered in the notes as well as several additional sections). While the notes below occasionally see corrections, all new development happens only in the just-mentioned text.
- Week 0:
- Week 1:
- Week 2:
- Week 3:
- Week 4:
- Week 5:
- Week 6:
- Week 7:
- Week 8:
- Week 9:
Grading and policies
- Grading matrix:
-
- Grade scale:
- These numbers are tentative and subject to change:
- A+, A, A-: (80%, 100%].
- B+, B, B-: (60%, 80%].
- C+, C, C-: (40%, 60%].
- D+, D, D-: (20%, 40%].
- Homework policy:
- Collaboration and reading is allowed, but you have to write solutions in your own words and acknowledge all sources that you used.
- Asking outsiders (anyone apart from Math 533 students and Drexel staff) for help with the problems is not allowed. (In particular, you cannot post homework as questions on math.stackexchange before the due date!)
- Late homework will not be accepted.
- Solutions have to be submitted electronically via Gradescope in a format I can read (PDF, TeX or plain text if it works; no doc/docx!). If you submit a PDF, make sure that it is readable and does not go over the margins. If there are problems with submission, send your work to me by email for good measure.
- Expected outcomes:
- The students should have an understanding of the basic objects of abstract algebra including rings and modules. They should also gain an understanding of homomorphisms, direct sums and products, and tensor products. They should have a knowledge of the basic theorems in this area including isomorphism theorems and universal properties. They should be familiar with the elementary properties of Gröbner bases and field extensions.