Math 101: Topics in Algebra

Fall 2016


Course Info:



We will follow the official Math 101 syllabus as closely as possible.

[PDF] Syllabus

Exercises, unless otherwise indicated, are out of Dummit and Foote.

Groups, first pass
112 Sep(M)Introduction; Some set theory;
selections from 1.1-1.6: Groups
WS 1 [TeX] [PDF]
213 Sep(T)1.3/3.5: Symmetric group; 1.7: Group actions1.7.21, 1.7.23
314 Sep(W)2.1-2.5: Subgroups;
3.1-3.5: Quotient groups and homomorphisms
WS 3 [TeX] [PDF]
Linear algebra
416 Sep(F)11.1: Definitions and basic theory 11.1.7
WS 4 [TeX] [PDF]
519 Sep(M)11.2: The matrix of a linear transformation11.2.11
620 Sep(T)Linear algebra extravaganzaWS 6 [TeX] [PDF]
721 Sep(W)11.3: Dual vector spaces, adjointsHW 7 [TeX] [PDF]
823 Sep(F)Tensor products over fields11.2.38 + what about det?,
926 Sep(M)Quadratic forms and bilinear formsHW 9 [TeX] [PDF] (updated 27 Sep)
1027 Sep(T)7.1-7.4: Review of rings; Rings extravaganza WS 10 [TeX] [PDF]
1128 Sep(W)11.4: Determinants; 11.5: Tensor, symmetric, and exterior algebras11.5.13
1230 Sep(F)7.5: Rings of fractions7.5.5
7.6.8-7.6.11 (due 11 Oct (T))
133 Oct(M)7.6: Chinese remainder (Sun Tsu) theorem
8.1: Euclidean domains
144 Oct(T)8.2: Principal ideal domains
8.3: Unique factorization domains
8.2.8, 8.3.5
155 Oct(W)9.1-9.4: Polynomial ringsHW 15 [TeX] [PDF]
167 Oct(F)10.1: Basic definitions and examplesHW 16 [TeX] [PDF]
1710 Oct(M)10.2: Quotient modules and module homomorphisms10.2.7 (due 12 Oct (W))
1811 Oct(T)Direct and inverse limitsHW 12 Solutions [TeX] [PDF]
-11 Oct
4:00-6:00 p.m.
(T)Midterm exam, covering the above through 5 Oct (W)Exam [TeX] [PDF]
Solutions [TeX] [PDF]
1912 Oct(W)10.3: Generation of modules, direct sums, and free modules10.3.2, HW 19 [TeX] [PDF]
2014 Oct(F)10.4: Tensor products of modulesHW 20 [TeX] [PDF]
2117 Oct(M)10.5: Exact sequences
2218 Oct(T)Hensel's lemma
2319 Oct(W)Diagram chases, splittingHW 23 [TeX] [PDF]
2421 Oct(F)10.5: Projective and injective modules10.5.8, 10.5.9
2524 Oct(M)15.4: Localization15.4.15
Category theory
2625 Oct(T)Appendix II: CategoriesHW 26 [TeX] [PDF]
2726 Oct(W)Appendix II: Functors, natural transformationsII.1.3, II.1.5
Modules over PIDs, canonical forms
2828 Oct(F)12.1: The basic theory12.1.2, 12.1.5
2931 Oct(M)12.2: Rational canonical form12.2.3, 12.2.4, 12.2.10,
12.3.2, 12.3.17, 12.3.22
(due 8 Nov (T))
-1 Nov(T)No class: JV at Fields Institute
-2 Nov(W)No class: JV at Fields Institute
Groups, second pass
304 Nov(F)12.3: Jordan canonical form
317 Nov(M)Smith normal formHW 31 [TeX] [PDF]
328 Nov(T)12.2, 12.3
339 Nov(W)4.1: Group actions and permutation representations;
4.3: Class equation
3.1.36, 4.3.6
3411 Nov(F)4.5: The Sylow theorems4.5.13, 4.5.31
3514 Nov(M)5.5: Semidirect products
3615 Nov(T)Wrap-upHomework self-assessment due
-22 Nov
8:00 a.m.
(T)Final exam, comprehensiveExam [TeX] [PDF]
Solutions [TeX] [PDF]



The homework assignments will be assigned on a daily basis and will be posted above. Homework is due the following class period: we will discuss the problem in class, and you will provide a self-assessment in red pen or pencil. At the end of the term, all homework will be collected, with a short concluding self-assessment.

Cooperation on homework is permitted (and encouraged), but if you work together, do not take any paper away with you--in other words, you can share your thoughts (say on a blackboard), but you have to walk away with only your understanding. In particular, you must write the solution up on your own. Please acknowledge any cooperative work at the end of each assignment.

Plagiarism, collusion, or other violations of the Academic Honor Principle, after consultation, will be referred to the The Committee on Standards.

[PDF] Homework Submission Guidelines