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-Holder 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.

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-Hoelder 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