| Prerequisites: |
Previous exposure to linear algebra (vectors, matrices, determinants)
is a must. One should either have the ability to write and read mathematical proofs, or have the desire and drive to learn how. |
| Instructor: | Victor Reiner (You can call me "Vic"). |
| Office: Vincent Hall 256 Telephone (with voice mail): 625-6682 E-mail: reiner@math.umn.edu |
|
| Classes: | Mon-Wed-Fri 10:10-11:00am, Vincent Hall 211. |
| Office hours: | Monday and Friday 9:05-9:55am, Tuesday 12:20-1:10pm, and by appointment. |
| Course content: |
This is the first semester in a 2-semester sequence on
the algebra of groups, ring, fields, and vector spaces. Roughly speaking the Fall and Spring semesters should divide the topics as follows: Fall-- Vector spaces, linear algebra, group theory and symmetry Spring-- Rings, modules, and field theory To give a feeling for the Fall subject matter, group theory can be thought of as the study of symmetry. Some nice examples of finite groups are the symmetries of regular polyhedra, like Platonic solids. |
| Required text: | Algebra, 2nd edition, by Michael Artin, Prentice-Hall, 2017.
The (very) tentative plan for proceeding through Artin's book goes like this: Fall-- some (but not all) of Chapters 1-8, very light on Chaps. 5,8 Spring-- some (but not all) of Chapters 11,12,14,15,16 |
| Level | Title | Author(s), Publ. info | Location |
|---|---|---|---|
| Lower (like Math 4281) |
Algebra: abstract and concrete | Goodman | author's download page |
| Same | Judson | Abstract algebra: theory and applications | author's download page |
| Topics in algebra | Herstein, Wiley & Sons | In Math Library (QA155.H4 1975b) | |
| Higher (like Math 8201) |
Abstract algebra | Dummit and Foote, Wiley & Sons | On reserve in math library |
| Algebra | Lang, Springer/Nature | On reserve in math library | |
| Proof writing and reading |
How to read and do proofs | Solow, Wiley & Sons | In Math Library (QA9.54.S65 2014) |
| How to prove it | Velleman | In Math Library (QA9.V38 1994) |
| Homework, exams, grading: |
There will be 5 homework assignments due Wednesdays,
usually every other week, but
I encourage collaboration on the homework, as long as each person understands the solutions, writes them up in their own words, and indicates on the homework page their collaborators. The take-home midterms and final exam are open-book, open-library, open-web, but in contrast to the homework on exams, no collaboration or consultation of human sources is allowed. Late homework will not be accepted. Early homework is fine, and can be left in my mailbox in the School of Math mailroom near Vincent Hall 105. Homework solutions should be well-explained-- the grader is told not to give credit for an unsupported answer. Complaints about the grading should be brought to me. |
| Final course grade basis : |
|
| Assignment or Exam | Due date | Problems from Artin, unless otherwise specified |
Lecture notes |
|---|---|---|---|
| Homework 1 | 9/19 |
1.1: 4,7,9,12ac,13 1.2: 2,5,8 1.3: 1,2 1.4: 1,4,6 1.5: 1,3,4 1.6: 2 Chap 1 Misc Probs: 3,4 |
Sept. 5 Sept. 7 Sept. 10 Sept. 12 Two answers to "Do permutations have some real-world application?": -- a talk on how permutations helped crack the Nazi's "Enigma" cipher, -- a paper giving a public-key cryptosystem based on the symmetric group Sn. Sept. 14 Sept. 17 |
| Homework 2 | 10/3 |
2.1: 3 2.2: 3,4 2.3: 1,2 2.4: 1,3,4,9,10 2.5: 1,2,3,4,6 2.6: 2,3,5,6,8 |
Sept. 19 Sept. 21 Sept. 24, P. Pylyavskyy subbing Sept. 26, D. Grinberg subbing Sept. 28, D. Grinberg subbing (+ 2 office hours: VinH 203B, Sept. 28, 9:05-9:55am, 12:20-1:10pm) Oct. 1, D. Grinberg subbing |
| Exam 1 | 10/10 | Midterm 1 | |
| Homework 3 | 10/24 |
2.8: 3,4,9,10 2.9: 2,4,5 2.10: 1 2.11: 4,6,9 2.12: 2 Chap 2 Misc Probs: 9,10 6.7: 1,7 6.8: 2,3 6.9: 1 6.10: 2 |
Oct. 3 Oct. 5 Oct. 8 Oct. 10 Oct. 12 Oct. 15 Oct. 17 Oct. 19, D. Grinberg subbing |
| Homework 4 | 11/7 |
7.1: 1 7.2: 2,16(a),17,7,18 (By 7.2.16(a), I mean skip the 2nd part asking you to show |C'| divides |C|, which is significantly trickier.) 7.3: 2 7.4: 7,9 7.5: 2,4,5,6 7.6: 1,2 |
Oct. 22 Identifying the composition of two rotations as a rotation. Oct. 24 Oct. 26 Oct. 29 Oct. 31 Nov. 2 |
| Exam 2 | 11/14 | Midterm 2 | |
| Homework 5 | 12/5 |
7.7: 3 7.8: 1 7.9: 1 7.10: 2,6,7 3.2: 1, 5, 6, 9 3.3: 2 3.4: 1, 2, 3, 6 3.5: 2, 4 Chap 3 Misc Probs: 1,2 |
Nov. 5 Nov. 7 Nov. 9 Nov. 12 Nov. 14 Nov. 16 Nov. 19 Nov. 21 Nov. 26 Nov. 28 Nov. 30 Dec. 3 Dec. 5 Dec. 7 Dec. 10 Dec. 12 |
| Final Exam | 12/12 | Final exam |