Math 410 - Module Theory and Commutative Algebra - Fall 2020

General Information:

Meeting Time:MWF 9 - 9:50
Location: Zoom/Jones 306
Instructor:Ryan Vinroot
Office Hours: All will be on Zoom, times TBD.
Textbook:Abstract Algebra (Third Edition) by David S. Dummit and Richard M. Foote
Grade
Breakdown: 		Class Participation - 10%, Homework - 60%, Final Exam - 30%. The grading scale will be the standard 10 percentage point scale, so that a final score of 93 or higher is an A, 90-92 is an A-, 87-89 is a B+, 83-86 is a B, 80-82 is a B-, 77-79 is a C+, 73-76 is a C, 70-72 is a C-, etc.
Attendance & Lecture Policy:It is expected that you attend all lectures (whether on Zoom or in the classroom), with exceptions minimized. It is greatly appreciated when you are on time. Please do your best to stay awake and attentive during lecture, please do not email or text during lecture, and keep all cell phones/hand held devices/tablets/laptops put away during lecture (unless you are specifically writing notes on a tablet). While it is understandable that you may miss a lecture here and there, or be sleepy in class once in awhile, repeated absences, late arrivals, naps, or general non-attentiveness will negatively affect your class participation score.
Prerequisite: Math 430 - Abstract Algebra II
Course Summary: The bulk of this course will be the material in Chapters 10, 12, and 15 of Dummit and Foote's text. We will begin with the basics of module theory given in Sections 10.1, 10.2, and 10.3, and then move on to the structure of modules over a PID in Chapter 12. We will then return to Section 10.4 to talk about tensor products of modules, followed by Section 11.5 on tensor algebras. This will hopefully take roughly half of the semester, at which point we will begin talking about commutative algebra in Chapter 15. We will see how far we get on that topic, and also consider going back to Section 10.5 to talk about exact sequences.

Dates & Course Announcements:

Exams:

There will be no midterm. The final exam will be a take-home exam, and due by the end of our final exam time slot, which is Fri, Nov. 20, 9 AM-12 Noon (so the take-home final will be due at noon that day).

Announcements:
  • All relevant announcements will be listed here. Check back frequently (don't forget to refresh your browser) for updates.
  • Important Dates and Class Holidays:
    • Fri, Aug 28: ADD/DROP DEADLINE
    • Mon, Oct 12: WITHDRAW DEADLINE
    • Fri, Nov 13: LAST DAY OF CLASS
    • Fri, Nov 20, 9:00 AM - 12 Noon - FINAL EXAM
  • (8/19) I will determine regular office hours after the add/drop period. I will not have any office hours during this short week, unless they are requested (which is welcomed).
  • (10/23) Apart from the 3 problems assigned for HW 10, there are also the two optional problems which came up in the notes from Day 28. There is no due date for these, just anytime by the end of the last day of classes.

Homework:

Homework problems will be a very important part of the course, and there will be homework assigned almost every week. Proofs and computations should be written carefully and neatly, with attention paid to the completeness of your argument and clarity of your steps. Individual homework assignments should be written up by yourself, although some collaboration while working on the homework is fine, and encouraged as long as the work you turn in is your own formulation of a solution. You should not, under any circumstances, attempt to copy solutions to problems online (although I know this is very tempting), as this will have to be treated as plagiarism. Instead, email me for a hint, or discuss problems in a group of classmates.

Homework problems will each be graded out of 10 points. Your homework score will be based on your individual problem scores, rather than your scores on assignments. In particular, there will be at least 50 problems assigned for homework throughout the semester. Your total Homework score for the course will be based on your best 45 problem scores, along with full credit for 5 other earnestly attempted problems which are turned in on time. Your homework counts for 60% of your grade, which breaks down as your best 50 scores out of 10 points each, plus full credit for 10 attempted problems.

Homework problems are due on the posted due date at the start of class, either turned in as a hard copy (if we move to the classroom), or emailed to me as a pdf of a latex file, or preferrably as just a latex file which I can then make comments on within the latex file directly. Homework is considered on time if it is turned in, or in my inbox, at most 10 minutes after the start of class.

A total of 10 Homework problems throughout the semester may be turned in up to 2 weekdays late (by 5 PM in my office or inbox) without penalty. No other late homework will be accepted for scoring (unless you have specific accommodations). All homework problems and their due dates will be listed below.
 
Assignment Problems Due Date
1 10.1 (pg. 344) #8(a), 9, 10
10.2 (pg. 350) #7, 8, 9
Fri, Aug 28
2 10.2 (pg. 350) #11, 12, 13
10.3 (pg. 356) #2, 6, 7
Fri, Sept 4
3 10.3 (pg. 357) #16, 17, 18
12.1 (pgs. 468-469) #2(a), 2(b), 3
Fri, Sept 11
4 10.3 #20(a), 20(b)
12.1 #4, 6, 8, 11
Fri, Sept 18
5 12.1 #15, 16
12.2 #4, 8, 10, 14
Fri, Sept 25
6 12.2 #18, 19
12.3 #17, 22, 23, 24
Fri, Oct 2
7 10.4 #3, 4, 5, 7, 16, 18
 Fri, Oct 9
8 10.4 #10, 11, 12, 21(a)
11.2 (pg. 431) #38, 39(a,b)
Fri, Oct 16
9 15.1 #1, 8, 14, 21
 Fri, Oct 23
10 15.1 #16, 23, 28 Fri, Oct 30
11 15.2 #2(c,d,e), 3, 7, 9, 10, 20 Fri, Nov 6

Math Major Writing Requirement (Math 300):

If you are a math major, and you would like to complete your major writing requirement through a writing assignment in this class, please let me know in the first week of class. This writing assignment will not count towards your grade in this class, but will rather just serve as your Major Writing Requirement (Math 300). You should only do this if all of the following hold: (1) you are not doing an honors thesis in Mathematics, (2) you are not doing your COLL 400 requirement in Mathematics, and (3) you are a senior. If you decide to do this, you must write your paper on a topic in Abstract Algebra (or a closely related subject) approved by me, and you must keep to a schedule of turning in drafts we agree on at the beginning of the semester in order to get credit.

Student Accessibility Services:

William & Mary accommodates students with disabilities in accordance with federal laws and university policy. Any student who feels they may need an accommodation based on the impact of a learning, psychiatric, physical, or chronic health diagnosis should contact Student Accessibility Services staff at 757-221-2512 or at sas@wm.edu to determine if accommodations are warranted and to obtain an official letter of accommodation. For more information, please visit the SAS webpage.