Math 410 - Galois Theory - Fall 2019

General Information:

Meeting Time:MWF 9 - 9:50
Location: Jones 113
Instructor:Ryan Vinroot
Office: Jones 100D
Office Hours: Mon 2-3, Wed 3-4, Thurs 10-11 and 2-3:30 (also by appointment).
Textbook:Abstract Algebra (Third Edition) by David S. Dummit and Richard M. Foote
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, 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 13 and 14 of Dummit and Foote's text. We will pick up where we left off in Math 430 with splitting fields, and we will in particular look at some detailed examples including cyclotomic fields. We will cover the rest of Chapter 13, including the existence and uniqueness of algebraic closures using Zorn's Lemma. We will then prove the Fundamental Theorem of Galois Theory in Chapter 14, and cover its applications to some important families of examples. Before proving the Abel-Ruffini Theorem on the insolvability of the quintic, we will take a necessary detour back to group theory to study properties of solvable groups. After completing Chapter 14 of the book, we may select some topics in the remaining chapters, depending on the remaining time in the semester.

Dates & Course Announcements:


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 Tues, Dec. 17, 9 AM-12 Noon (so the take-home final will be due at noon that day).

  • All relevant announcements will be listed here. Check back frequently (don't forget to refresh your browser) for updates.
  • Important Dates and Class Holidays:
    • Mon, Sep 6: ADD/DROP DEADLINE
    • Sat, Oct 12 - Sun, Oct 15: NO CLASS (Fall Break)
    • Fri, Oct 25: WITHDRAW DEADLINE
    • Wed, Nov 27 - Sun, Dec 1: NO CLASS (Thanksgiving Break)
    • Fri, Dec 6 - LAST DAY OF CLASS
    • Tues, Dec 17, 9:00 AM - 12 Noon - FINAL EXAM
  • (8/28) I will determine regular office hours after the add/drop period. For this short week, my office hours will be: Wed Aug 28 2-3:30, Thurs Aug 29 10:30-12 and 2:30-4.
  • (9/2) The first HW has been posted. My office hours this week will be: Mon 2-3, Wed 3-4, Thurs 10:30-11:30 and 2:30-3:30.
  • (9/9) My office hours this week, which might end up being my standard weekly office hours, will be: Mon 2-3, Wed 3-4, Thurs 10-11 and 2-3:30.
  • (9/13) I will give the first optional half lecture on Mon morning, Sept 16, at 8:30 am in our regular room. I will start the discussion on the equivalence of the Axiom of Choice, Zorn's Lemma, and the Well-Ordering Principle.
  • (9/16) My office hours last week will become my permanent weekly office hours: Mon 2-3, Wed 3-4, Thurs 10-11 and 2-3:30.
  • (10/9) I have to cancel my morning office hours tomorrow morning, Thurs Oct 10, 10-11 am. I will still have my afternoon office hours tomorrow 2-3:30 pm.
  • (10/11) As promised, here is some optional homework based on the optional half-lectures, all of which could be done just using the notes that I followed (so you can still work on these even if you did not come to the optional half lectures): Optional HW. You may turn these in for credit (either for a counted score, or a counted attempted problem) any time before 5 PM on the last day of classes this semester.
  • (11/4) I will be out of town later this week, and so my office hours will be different this week. They are: Mon Nov 4 2-3:30, Tues Nov 5 1:30-3, and Wed Nov 6 2:30-4.
    Also: This Fri, Nov 8, you all will be running class yourselves. Your assignment as a class is to get through the material in Sec. 14.6 under "Polynomials of Degree 4" starting on pg. 613, through pg. 615 up to the fundamental theorem of algebra (which we've already covered). One of your HW scores this week will be given for showing up to class on Friday and participating in going through this material with your classmates. A moderator (sub) will be there to make sure you are there and are going through material, so class still starts at 9 AM on Friday. You should all go through the material before hand, and you can break up the presentation of the material (use the chalkboard!) any way you want.
  • (11/11) The following are the notes of Keith Conrad that I was using for generating sets of Sn, I was using Section 2: Notes on generating sets. I also proved a statement that was not in these notes which we will need: Any transitive subgroup of Sn containing a transposition and an (n-1)-cycle is all of Sn, which we proved from Corollary 2.6 in the linked notes.
  • (11/15) Today in the optional half lecture, we are finishing the proof of the fundamental theorem of finite abelian groups. As optional problems (due no later than the last day of classes) I am assigning 13.6, pg. 557 #14, 15, 16, 17. These problems give a proof that for any m>1, there are infinitely many primes congruent to 1 mod m. These facts give enough for the complete proof in Sec. 14.5 that any finite abelian group is the Galois group of some extension of Q contained in a cyclotomic field.
  • (11/18) My office hours this week will be as follows: Mon (Nov 18) 2-3:30, Wed 12:30-2, Fri 3-4:30.
  • (11/25) Here is a video from the Mathologer YouTube channel about Cardano's cubic formula. Since we are not covering this in Sec. 14.7 of the book, you should watch this video, which also has some nice history in it: The Cubic Formula.
  • (11/25) Here are 4 more optional problems, which are due any time by the beginning of the last day of class: pgs. 637-638 #13, 16, 17, 18.
  • (12/6) The take-home final is being given out in class today, and will be due by 12 Noon on Tues, Dec 17. I will be available in my office during exams from 10 AM-12 Noon and 2-3:30 PM on Mon Dec 9, Wed Dec 11, Thurs Dec 12, Fri Dec 13, and Mon Dec 16.


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 60 problems assigned for homework throughout the semester. Your total Homework score for the course will be based on your best 50 problem scores, along with full credit for 10 other earnestly attempted problems which are turned in on time. That is, 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, or emailed to me as a pdf of a latex file (scanned homework is not accepted). 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 pg. 545 #2, 4
pg. 551 #1 (just product), 3, 5, 6
Mon, Sept 9
2 pgs. 551-552 #4, 11
pgs. 555-556 #5, 6, 13(a,b), pg. 259 #35
Mon, Sept 16
3 pgs. 556-557 #13(c,d)
pgs. 566-567 #1(a), 4, 5, 6
Mon, Sept 23
4 pg. 557 #13(e)
pgs. 566-567 #1(b), 7, 10
Mon, Sept 30
5 pgs. 581-582 #1, 2, 12, 13, 14 Mon, Oct 7
6 pgs. 582-583 #8, 16, 17, 18
pg. 589 #1, 8
Mon, Oct 21
7 pg. 582 #15, pg. 589 #6, pg. 596 #5(a)
pgs. 603-604 #5, 10, 13
Mon, Oct 28
8 pg. 603 #7, 8, pgs. 617-621 #4, 17, 21, 37 Mon, Nov 4
9 pgs. 617-622 #12, 13, 14, 38
Go through material on Friday
in class for a HW problem credit.
Mon, Nov 11
10pgs. 617-623 #3, 18, 22, 39, 46, 47 Mon, Nov 18
11pg. 106 #5, pg. 623 #44
pgs. 636-637 #4, 7, 12
Mon, Nov 25

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 to determine if accommodations are warranted and to obtain an official letter of accommodation. For more information, please visit the SAS webpage.