## 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 004 201

• 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 7.5.57.6.8-7.6.11 (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 Oct4:00-6:00 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 Nov8:00 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