MAT 1100: Algebra I, Fall 2024
Instructor: Florian Herzig;
my last name at math dot toronto dot edu
Office Hours (online/zoom): Mon 2-3 (see quercus).
TA: Nischay Reddy
Lectures: Tuesdays/Thursdays 11:30am-1pm
Syllabus
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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.
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Ring Theory: Rings, ideals, Euclidean domains, principal ideal domains, unique factorization domains, field of fractions.
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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)
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: 25%
Term test: 30%
Final: 45%
There will be about 5 homework assignments. Your lowest homework score will not count towards your grade.
Drop deadline: October 28, 2024
Term test (tentative): Tue, Oct 22, 11:30am-1:00pm (instead of class).
There will be no makeup test! If you miss the test for a valid reason, the grade will be reweighted as 36% homework and 64% final.
Final: TBA (in person)
Homework (tentative schedule)
Assignments will be posted and solutions will be collected via Gradescope.
- Assignment 1, due Wed Oct 2
- Assignment 2, due Wed Oct 16
- Assignment 3, due Wed Oct 30
- Assignment 4, due Wed Nov 20
- Assignment 5, due Wed Dec 4
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!
Rough class schedule
- Sep 3: Group theory: basic definitions and examples, subgroups, homo/iso/automorphisms, kernel/image
- Sep 5: group actions (orbits, stabilisers), cosets, Cayley's theorem
- Sep 10: normal subgroups, quotient group and homomorphism, universal property of quotient groups
- Sep 12: isomorphism theorems
- Sep 17: direct products, cyclic groups, symmetric groups
- Sep 19: simplicity of A_n, Sylow theorems
- Sep 24: Sylow theorems and applications
- Sep 26: no class
- Oct 1: semidirect products, JH theorem
- Oct 3: JH theorem, solvable groups
- Oct 8: free groups
- Oct 10: free groups, presentations; Ring theory: definitions and first examples
- Oct 11 (make-up class): subrings, ring homomorphisms, quotient rings, isomorphism theorems
- Oct 15: direct products, fields/integral domains, maximal/prime ideals, existence of maximal ideals
- Oct 17: no class
- Oct 22: Term Test
- Oct 24: prime and irreducible elements, PIDs and UFDs, an example of a non-UFD
- Oct 29: noetherian rings, PID implies UFD, Euclidean domains, gcd
- Oct 31: field of fractions, content/primitive polynomials, Gauss lemma (both forms), start of R UFD implies R[x] UFD
- Nov 5: R UFD implies R[x] UFD, Eisenstein criterion, Module theory: examples, homomorphisms, submodules
- Nov 7: quotient modules and isomorphism theorems, direct products/direct sums, free modules and bases, motivation for tensor products
- Nov 12: tensor products
- Nov 14: extension of scalars, f.g. modules over a PID (statements of the two canonical forms)
- Nov 19: part of proof of canonical forms (every such module is direct sum of cyclic modules); chinese remainder theorem
- Nov 21: finished proof of canonical forms
- Nov 26: applications to linear algebra: minimal/characteristic polys, companion matrices, rational canonical form
- Nov 28: rational canonical form, Jordan canonical form
Links