Math 1100 - Algebra I, Fall 2023
Instructor: Daniel Litt; daniel.litt [at] utoronto [dot] ca
Office Hours: Thursdays 3-4 or by appointment at 215 Huron, Office 1028
Location: W 11-12, Th 11-1, BA6183
Syllabus
Category theory: Universal properties, adjoint functors, the Yoneda lemma
Group Theory: Isomorphism theorems, group actions, Jordan–Hölder 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 loosely follow Aluffi and Grillet)
Aluffi: Algebra, Chapter 0
P.A. Grillet, Abstract Algebra, 2007.
Lang, Algebra, 3rd ed.
Dummit and Foote, Abstract Algebra, 3rd ed.
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 19, 11:10-1:00 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 December 13 from 9am-12 noon in BA 6183.
Homework (tentative schedule)
The solutions for these homeworks will be collected via Crowdmark.
Assignment 1: Due September 28
Assignment 2: Due October 12
Assignment 3: Due October 26
Assignment 4: Due November 16
Assignment 5: Due November 30
Weekly plan
Week 1: Basics of category theory; definitions of groups, subgroups, homomorphisms; and rings; many examples
Week 2: quotients, isomorphism theorems, direct products, cyclic groups, symmetric groups, simple groups
Week 3: simplicity of A_n, group actions, orbit-stabiliser theorem, Cayley's theorem, Sylow theorems
Week 4: semidirect products, Jordan–Hölder theorem, derived subgroup
Week 5 solvable groups, free groups, presentations
Week 6: Thanksgiving and Term Test
Week 7: ring theory: subrings, homomorphisms, ideals, isomorphism theorems
Week 8: maximal and prime ideals, PIDs and UFDs, Euclidean domains, gcd
Week 9: field of fractions, Gauss' lemma, Eisenstein criterion
Week 10: modules, sub/quotient modules, homomorphisms, direct sums/products, tensor products
Week 11: extension of scalars, classification of finitely generated modules over PIDs
Week 12: classification of finitely generated modules over PIDs, applications to linear algebra