Florian Herzig - Math 1100

MAT 1100: Algebra I, Fall 2019

Instructor: Florian Herzig; my last name at math dot toronto dot edu
Office Hours: Thursdays 3:30-4:30 or by appointment at BA 6186

TA: Daniel Spivak

Lectures: Mondays 10-11am, Thursdays 1-3pm at BA 6183

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)

Lang, Algebra, 3rd ed.
Dummit and Foote, Abstract Algebra, 3rd ed.

Also recommended:

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: 25%
Term test: 25%
Final: 50%

There will be about 5 homework assignments. Your lowest homework score will not count towards your grade.
The term test will be on Thu, Oct 17, 1:10-3:00pm in BA6183 (instead of class). There will be no makeup test! If you miss the test for a valid reason, the grade will be reweighted as 35% homework and 65% final.
The final will be on Wed, Dec 11, 1:10-4:00pm in BA6183.

Homework (tentative schedule)

The solutions for these homeworks will be collected via Crowdmark.

Assignment 1, due Tue Oct 1
Assignment 2, due Tue Oct 15
Assignment 3, due Tue Oct 29
Assignment 4, due Tue Nov 19
Assignment 5, due Fri Dec 6

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!

Weekly plan

Week of Sep 9: review of groups, subgroups, homomorphisms; normal subgroups and quotient groups (+universal property)
Week of Sep 16: isomorphism theorems, direct products, cyclic groups, symmetric groups, simple groups
Week of Sep 23: simplicity of A_n, group actions, orbit-stabiliser theorem, Cayley's theorem, Sylow theorems
Week of Sep 30: semidirect products, Jordan-Hoelder theorem, derived subgroup
Week of Oct 7: solvable groups, free groups, presentations
Week of Oct 14: Thanksgiving and Term Test
Week of Oct 21: I will be away, class will be made up later
Week of Oct 28: ring theory: subrings, homomorphisms, ideals, isomorphism theorems
Week of Nov 4: maximal and prime ideals, PIDs and UFDs, Euclidean domains, gcd
- schedule for reading week: Mon 10-11am (Nov 4) and Wed 3-5pm (Nov 6) at BA6183
Week of Nov 11: field of fractions, Gauss' lemma, Eisenstein criterion
Week of Nov 18: modules, sub/quotient modules, homomorphisms, direct sums/products, tensor products
Week of Nov 25: extension of scalars, classification of finitely generated modules over PIDs
Week of Dec 2: classification of finitely generated modules over PIDs, applications to linear algebra
- there will be a make-up class on Thu Dec 5, 10am-12pm

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)
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.)
Proof of the elementary divisor theorem. (See Theorem 7.1 and its proof in these notes.)