Math 101: Topics in Algebra
Fall 2016
Course Info:
- Lectures: Monday, Wednesday, Friday, block 12 (12:50 p.m. - 1:55 p.m.)
- x-period: Tuesday, block 12X (1:20 - 2:10 p.m.)
- Dates: 12 September 2016 - 14 November 2016
- Room: Kemeny
004201 - Instructor: John Voight
- Office: 341 Kemeny Hall
- E-mail: jvoight@gmail.com
- Office hours: Thursday, 3:30-4:30 p.m., as well as Monday 3:30-4:30 p.m. and Tuesday 4:00-6:00, or by appointment
- Course Web Page: http://www.math.dartmouth.edu/~m101f16/
- Prerequisites: A previous course in undergraduate algebra is strongly recommended.
- Required Text: David S. Dummit and Richard M. Foote, Abstract Algebra, 3rd ed., Wiley, 2003; see their errata.
- Recommended Texts:
- Serge Lang, Algebra, 3rd. ed., GTM vol. 211, Springer-Verlag, 2005.
- Thomas Hungerford, Algebra, 8th ed., GTM vol. 73, Springer-Verlag, 2003.
- Paolo Aluffi, Algebra: Chapter 0, GSM, American Mathematical Society, 2009.
- Michael Artin, Algebra, 2nd. ed., Pearson, 2010.
- Grading: Grade will be based on daily homework (50%), a midterm (20%), and a final exam (30%).
Syllabus:
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 | 1 | 12 Sep | (M) | Introduction; Some set theory; selections from 1.1-1.6: Groups | 1.4.11 WS 1 [TeX] [PDF] |
| 2 | 13 Sep | (T) | 1.3/3.5: Symmetric group; 1.7: Group actions | 1.7.21, 1.7.23 |
| 3 | 14 Sep | (W) | 2.1-2.5: Subgroups; 3.1-3.5: Quotient groups and homomorphisms | 3.2.14 WS 3 [TeX] [PDF] |
| Linear algebra | ||||
| 4 | 16 Sep | (F) | 11.1: Definitions and basic theory | 11.1.7 WS 4 [TeX] [PDF] |
| 5 | 19 Sep | (M) | 11.2: The matrix of a linear transformation | 11.2.11 |
| 6 | 20 Sep | (T) | Linear algebra extravaganza | WS 6 [TeX] [PDF] |
| 7 | 21 Sep | (W) | 11.3: Dual vector spaces, adjoints | HW 7 [TeX] [PDF] |
| 8 | 23 Sep | (F) | Tensor products over fields | 11.2.38 + what about det?, 11.2.39 |
| 9 | 26 Sep | (M) | Quadratic forms and bilinear forms | HW 9 [TeX] [PDF] (updated 27 Sep) |
| Rings | ||||
| 10 | 27 Sep | (T) | 7.1-7.4: Review of rings; Rings extravaganza | WS 10 [TeX] [PDF] |
| 11 | 28 Sep | (W) | 11.4: Determinants; 11.5: Tensor, symmetric, and exterior algebras | 11.5.13 |
| 12 | 30 Sep | (F) | 7.5: Rings of fractions | 7.5.5 7.6.8-7.6.11 (due 11 Oct (T)) |
| 13 | 3 Oct | (M) | 7.6: Chinese remainder (Sun Tsu) theorem 8.1: Euclidean domains | 8.1.7 |
| 14 | 4 Oct | (T) | 8.2: Principal ideal domains 8.3: Unique factorization domains | 8.2.8, 8.3.5 |
| 15 | 5 Oct | (W) | 9.1-9.4: Polynomial rings | HW 15 [TeX] [PDF] |
| Modules | ||||
| 16 | 7 Oct | (F) | 10.1: Basic definitions and examples | HW 16 [TeX] [PDF] |
| 17 | 10 Oct | (M) | 10.2: Quotient modules and module homomorphisms | 10.2.7 (due 12 Oct (W)) |
| 18 | 11 Oct | (T) | Direct and inverse limits | HW 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] |
| 19 | 12 Oct | (W) | 10.3: Generation of modules, direct sums, and free modules | 10.3.2, HW 19 [TeX] [PDF] |
| 20 | 14 Oct | (F) | 10.4: Tensor products of modules | HW 20 [TeX] [PDF] |
| 21 | 17 Oct | (M) | 10.5: Exact sequences | |
| 22 | 18 Oct | (T) | Hensel's lemma | |
| 23 | 19 Oct | (W) | Diagram chases, splitting | HW 23 [TeX] [PDF] |
| 24 | 21 Oct | (F) | 10.5: Projective and injective modules | 10.5.8, 10.5.9 |
| 25 | 24 Oct | (M) | 15.4: Localization | 15.4.15 |
| Category theory | ||||
| 26 | 25 Oct | (T) | Appendix II: Categories | HW 26 [TeX] [PDF] |
| 27 | 26 Oct | (W) | Appendix II: Functors, natural transformations | II.1.3, II.1.5 |
| Modules over PIDs, canonical forms | ||||
| 28 | 28 Oct | (F) | 12.1: The basic theory | 12.1.2, 12.1.5 |
| 29 | 31 Oct | (M) | 12.2: Rational canonical form | 12.2.3, 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 | ||||
| 30 | 4 Nov | (F) | 12.3: Jordan canonical form | |
| 31 | 7 Nov | (M) | Smith normal form | HW 31 [TeX] [PDF] |
| 32 | 8 Nov | (T) | 12.2, 12.3 | |
| 33 | 9 Nov | (W) | 4.1: Group actions and permutation representations; 4.3: Class equation | 3.1.36, 4.3.6 |
| 34 | 11 Nov | (F) | 4.5: The Sylow theorems | 4.5.13, 4.5.31 |
| 35 | 14 Nov | (M) | 5.5: Semidirect products | |
| 36 | 15 Nov | (T) | Wrap-up | Homework self-assessment due |
| - | 22 Nov 8:00 a.m. | (T) | Final exam, comprehensive | Exam [TeX] [PDF] Solutions [TeX] [PDF] |
Homework:
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