MAT 347: Groups, Rings, and Fields; 2026-2027
Instructor: Florian Herzig;
my last name at math dot toronto dot edu
Office Hours (online/zoom): Mon 3:30-4:30pm or by appointment
TA: Aleksandr Popkovich
TA: Nischay Reddy
Official syllabus
Lectures: Thursdays 1-3pm, Fridays 10-11am
Textbook: Abstract algebra by Dummit and Foote (Wiley, 3rd ed.)
Possible helpful books (not to replace the textbook): Aluffi's Algebra: Notes from the Underground; Artin's Algebra
Final assessment: during April examination period
Term tests: November 5 + January 21 + March 11, 1:10-3:00pm (during class time)
Course description:
This course will provide a detailed introduction to modern abstract algebra, which is a basic part of the language of much of modern mathematics. A highlight of this course will be Galois theory, one of the most beautiful gems of mathematics that you will learn as an undergraduate (which Galois developed at the age of 17, four years before dying in a duel). To get there, first we need to develop a lot of background: group theory, ring theory, and field theory. By the end of this course we will see interesting applications of Galois theory, such as why a regular heptagon cannot be constructed by straightedge and compass, and why a general quintic polynomial equation cannot be solved using radicals.
Warning: this course will be tough and very abstract. It is aimed at Mathematics Specialists. Do not take this class unless you are willing to put in a significant intellectual effort! But if you do put in the effort every week, then you will be rewarded by learning beautiful mathematics.
General tips for this course:
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Do not fall behind! The material will develop quickly in this course.
- Keep engaging with the course material by attending lectures and tutorials, reading the book, and (importantly) doing problems.
- LLMs can be very helpful as a tool, but real learning takes time and involves some struggle for most of us (I'm no exception)!
Homework and quizzes:
- There will be a quiz in tutorial, roughly every two weeks. Quizzes will help you get regular feedback.
- Homework will be assigned regularly and collected via gradescope. Assignments will not carry much credit, but will be a very useful way to practice for quizzes and exams.
Some quiz and test problems will be related to the assignments.
- You are encouraged to work together with other students on assignments! You are still expected to submit your own work.
Marking scheme:
- Term Tests (3): 45%
- Final assessment: 30%
- Quizzes: 20%
- Homework: 5%
There are no make-up quizzes and tests. The lowest three quiz scores will be dropped.
(The marking scheme will be reweighted if you provide a valid reason for missed term work within one week.)
Academic integrity:
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Please familiarise yourself with the University of Toronto Code of Behaviour on Academic Matters. See also a simplified version.
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The University of Toronto treats cases of academic misconduct very seriously. All suspected cases of academic dishonesty will be investigated following the procedures outlined in the Code. The consequences for academic misconduct can be severe, including a failure in the course and a notation on your transcript. Every year, students get expelled permanently for academic offences.
Accessibility:
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University of Toronto is committed to accessibility. If you require accommodations, or have any accessibility concerns about
the course, please contact Accessibility Services as soon as possible.
Week 1 (starting Sep 7):
- Reading:
- section 0.1 (useful for review; we will talk about equivalence relations in class)
- section 0.2 (quick review, we'll need this more later)
- sections 0.3, 1.1-1.3
- Why groups? (by Keith Conrad)
- Equivalence relations (see the first three sections; these notes go into slightly more detail than the textbook)
- Cayley's 1854 paper in
which he introduces finite groups, see the top of p.41 (though he thinks of elements as operations, so his "axioms" look a little different from ours, e.g. associativity is automatic for him). He also classifies groups of order 4 and 6.
Week 2 (starting Sep 14):
- Reading: sections 1.3-1.7.
- Quiz 1 in Friday's Tutorial.
- In class, we discussed a presentation of the Dihedral group with two generators, each of which squared to 1. This is a special case of a Coxeter group, named after Toronto's famous mathematician H.S.M. Coxeter (1907-2003, at U of T 1936-1996).
Week 3 (starting Sep 21):
- Reading: sections 1.7, 4.1, 2.1, 2.2, 2.3.
- Hwk 1 due on Friday.
Week 4 (starting Sep 28):
Week 5 (starting Oct 5):
- Reading: sections 3.3, 3.4, 3.5.
- Hwk 2 due on Friday.
