Florian Herzig - Math 1100

MAT 1100: Algebra I, Fall 2020

Instructor: Florian Herzig; my last name at math dot toronto dot edu
Office Hours (online/zoom): by appointment

TA: Pavel Shlykov

Lectures: Tuesdays 3-5pm, Thursdays 3-4pm (online/zoom)

Syllabus

Group Theory: Isomorphism theorems, group actions, Jordan-Hoelder theorem, Sylow theorems, direct and semidirect products, finitely generated abelian groups, simple groups, symmetric groups, linear groups, solvable groups, free groups, generators and relations.
Ring Theory: Rings, ideals, Euclidean domains, principal ideal domains, unique factorization domains, field of fractions.
Modules: Modules, tensor products, modules over a principal ideal domain, applications to linear algebra.

Useful books for reference (I will not really follow any of them)

Grillet, Abstract algebra (click link for online access; this book might be the closest to my course)
Lang, Algebra, 3rd ed.

Also recommended:

Dummit and Foote, Abstract Algebra, 3rd ed.
Paul Selick's course notes: Groups, Rings (from around 2011)
Dror Bar Natan's course page (2014)
Rotman, Robinson both in the GTM series (for groups; they cover much more!)

Grading scheme

Homework: 30%

Term test: 30%

Final: 40%

There will be about 5 homework assignments. Your lowest homework score will not count towards your grade.

Drop deadline: October 26, 2020

Term test: Tue, Oct 20, 3:10-5:00pm (instead of class). There will be no makeup test! If you miss the test for a valid reason, the grade will be reweighted as 45% homework and 55% final.
Final: Mon, Dec 21, 12:00-5:10pm (on gradescope; you will get 190 minutes)

Homework (tentative schedule)

Assignments will be posted and solutions will be collected via Gradescope.

Assignment 1, due Thu Oct 1
Assignment 2, due Fri Oct 16
Assignment 3, due Fri Oct 30
Assignment 4, due Fri Nov 20
Assignment 5, due Wed Dec 9

More problems, for practice: see all problems, mostly from Dummit-Foote, I assigned for my undergraduate algebra course!

My ring theory conventions

Please consult this note, especially if you use Dummit-Foote!

Links

George Bergman's companion to Lang's Algebra
Notes on universal property of the quotient group (you can just start reading at Definition 9.4; in particular, Theorem 9.5 gives the full proof that quotient groups are uniquely characterised by the universal property; the one thing I would change is just demand that the kernel of u contains H rather than be equal to H in Definition 9.4. It's then a consequence of the universal property that ker(u) is equal to H.)
Handout by Dror Bar Natan on simplicity of A_n (n \ge 5)
If you are interested in picking up the basics of categories, I suggest looking at the beginning of Chapter 16 in Grillet's book or the beginning of these notes.
An example of a PID that is not Euclidean. (There is a much nicer proof that this ring is a PID, but it requires some algebraic number theory.)
An elementary proof that many quadratic rings are UFDs.
Every vector space has a basis and any two bases have the same cardinality, using Zorn's lemma.