Math 171: Abstract Algebra

Fall 2018

Overview

Topics in group theory include groups, subgroups, quotient groups, Lagrange's theorem, symmetry groups, and the isomorphism theorems. Topics in Ring theory include Euclidean domains, PIDs, UFDs, fields, polynomial rings, ideal theory, and the isomorphism theorems. In recent years, additional topics have included the Sylow theorems, group actions, modules, representations, and introductory category theory.

Instructor

Dagan Karp
Office: 3414 Shan
Office hours: Tue 3-4pm, and open door.

Lectures

This course will meet for lectures Monday and Wednesday, 1:15-2:30pm in Shanahan 3421.

Textbook

We'll use Abstract Algebra: Theory and Applications, by Tom Judson.
It is freely available online, in both html and pdf.

Grading

There will be two exams and weekly homework. Let A be your homework average, B your score on Exam 1, C = Exam 2, D = Max (A,B,C) and Then your grade will be determined by
AVG (A,B,C,D)

Homework

Written homework will be due each Wednesday at the beginning of class, and is posted below.

You are encouraged to discuss the homework with other members of the class, and it is appropriate to acknowledge the assistance of others. You will, however, be expected to write up your solutions on your own. The instructor will reserve the right to refuse to accept late homework for any reason. Graded homework, however, may (in most cases) be rewritten and submitted at a later date. Please consult the HMC mathematics homework format guidelines for helpful tips on homework submission and formatting.

Rewrites

Like other writing, good math writing is a process that includes revision. In Math 171 this is reflected in the rewrite policy. You will have the opportunity to turn in a rewritten copy of every homework assignment.
For details, see the Math 171 Rewrite Policy.

Grutors

Feiyand Lin and Princewill Okoroafor are our graders and tutors. They will hold tutoring hours Sunday and Tuesday, 7-9pm in Shan 2444.

Exams

There will be two exams, both of which are take-home. Exam 1 will be distributed Wednesday, 10/3 and due 10/10. Exam 2 will be distributed Wednesday, 12/12, due Tuesday 12/18.

Critical Readings

In addition to the written homework, suggested readings will be posted on the homework page in conjunction with upcoming lectures. Before each lecture, read the corresponding material. Class activities will depend on your study of this material in advance (which is a departure from most lower division courses).

The goals of the critical reading exercises are manifold: to better the student's independent intake of mathematical exposition, to train in independent learning, to increase the interactive nature of the course (by allowing the instructor to respond to questions and comments), and to make the course more tailored to the specific curiosities of the class as a whole.

LaTeX

Students interested in using LaTeX are encouraged to do so, but it is not required.

Disabilities

It is the policy of The Claremont Colleges to accommodate students with temporary or permanent disabilities. Any student with a documented disability who requires reasonable accommodations should contact Deborah Kahn, Coordinator for Student Disability Resources at (909) 607-3148 or dkahn@hmc.edu, as soon as possible. Students from the other Claremont Colleges should contact their home college's disability officer. Counterparts at the other campuses are as follows: CMC: Julia Easley (julia.easley@cmc.edu) CGU: Chris Bass (chris.bass@cgu.edu) HMC: Deborah Kahn (dkahn@hmc.edu) Pitzer: Gabriella Tempestoso (gabriella_tempestoso@pitzer.edu) Pomona: Jan Collins-Eaglin (jan.collins-eaglin@pomona.edu) Scripps: Leslie Schnyder (lschnyde@scrippscollege.edu)

Math is for all

My goal is to welcome everyone to mathematics, and to broaden participation in the mathematical sciences. Our classroom should be an inclusive space, where ideas, questions, and misconceptions can be discussed with respect. There is usually more than one way to see and solve a problem and we will all be richer if we can be open to multiple paths to knowledge. I look forward to getting to know you all, as individuals and as a learning community.

Lecture Notes

Here are (old) course notes.
  1. Introduction to groups: symmetry and permutation
  2. Formal definition of group, first examples and properties
  3. Subgroups
  4. Cyclic Groups
  5. Symmetric and Dihedral Groups
  6. Cosets and Lagrange's Theorem
  7. Group Homomorphisms
  8. Normal Subgroups
  9. Isomorphism Theorems
  10. The Fundamental Theorem of Finitely Generated Abelian Groups
  11. Definition of Ring, first examples and properties
  12. Ring homomorphisms, subrings, and quotient rings
  13. Polynomial rings, matrix rings and group rings
  14. Ideals
  15. Maximal and Prime Ideals
  16. Euclidean Domains
  17. Principal Ideal Domains
  18. Unique Factorization Domains
  19. Polynomial Rings Revisited
  20. Introduction to Category Theory: Objects and Morphisms
  21. Functors

Homework and Readings
    HW 2. Due Wed Sep 19
    Reading: Chapter 4
    Problems (3.4) 40, 41, 42, 43, 45, 50.
    HW 3. Due Wed Sep 26
    Reading: Chapter 5
    Problems:(4.4) 4, 5, 36. (5.3) 1, 2, 6.
    Rewrite: HW 1
    HW 4. Due Wed Oct 3
    Reading: Chapter 6
    Problems:(6.4) 5 a-d, 11, 12.
    Rewrite: HW 2
    Exam 1
    Distributed: Tue Wed Oct 3
    Due: Wed Oct 10 (start of class)
    No homework due Wed, Feb 10.
    HW 5. Due Wed Oct 17
    Reading: Chapter 10 and 11
    Problems: (10.3) 5, 6, 14. (11.3) 4, 11, 19.
    Rewrite: HW 3
    No Class Mon. Oct 22: Fall Break
    HW 6. Due Wed Oct 24 (half assignment for Fall Break)
    Reading: Chapter 9
    Problems: (9.3) 23, 28, 49.
    Rewrite: HW 4
    HW 7. Due Wed Oct 31 (boo)
    Reading: Chapter 16
    Problems: (16.6) 1 (a-d), 1 (e-h), 2, 3a, 3e, 4.
    Rewrite: HW 5
    HW 8. Due Wed Nov 7
    Reading: Chapter 17
    Problems: (16.6) 6, 7, 29, 34, 35, 37, 38.
    Rewrite: HW 6.
    HW 9. Due Wed Nov 14
    Reading: Chapter 18.1
    Problems: (17.4) 1, 7, 10, 12, 26, 27.
    Rewrite: HW 7.
    HW 10. Due Monday November 19 (half assignment for Thanksgiving Break)
    Reading: Chapter 18.2
    Problems: (18.3) 11 (a) and (b), 14, 15.
    Rewrite: HW 8.
    No Class Wed Oct 21: Thanksgiving
    HW 11. Due Wed Dec 5
    Reading: Section 1.1 of Category Theory in Context by Emily Riehl.
    Problems: (18.3) 2, 5, 12, 13, 16, 19.
    Rewrite: HW 9.
    Final Project. Applications of Algebra.
    Research a particular application of abstract algebra in another field of mathematics (e.g., topology) or another discipline (e.g., chemistry).

    Due Mon Dec 10: Oral Presentation. Give a 3 minute oral presentation on your chosen application of algebra. Your talk should use roughly 3 slides. Submit your slides to your dropbox in Sakai by 10am on Monday, December 10.

    Due Wed Dec 12: Written exposition. Write a 1-2 page paper describing an application of algebra. Your piece should be written as though it could be included seamlessly in a future edition of our text book, and you must use at least one outside source (aside from wikipedia). Your paper must be typset in latex.

    Final Exam

    The second and final exam will be distributed on Wednesday, December 12 and due Tuesday, December 18 by 5pm.