Math 430 - Abstract Algebra II - Spring 2013

General Information:

Meeting Time:MWF, 11:00 - 11:50
Location: Jones 306
Instructor:Ryan Vinroot
Office: Jones 130
Office Hours: M 1:30-2:30, W 2:30-3:30, Th 9:30-10:30 AM and 3:30-5 PM, also by appointment/walk-in.
Textbook:A First Course in Abstract Algebra, Seventh Edition, by John B. Fraleigh
Class Participation - 5%, Midterm Exam - 30%, Homework - 30%, Final Exam - 35%. 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-, 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/laptops put away during lecture. 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 307 - Abstract Algebra I.
Syllabus: We will be covering topics in Groups, Rings, and Fields, which extend the concepts covered in Math 307. We will use the text as the main resource, with a few exceptions, but we will skip around in the book quite a bit. We will begin by covering the group isomorphism theorems in Section 34, and then cover Section 16 on Group Actions, followed by Sylow Theorems in Sections 36 and 37. We will then move to Rings, and quickly review some of the topics in Sections 22, 23, 26, and 27 which were mostly covered in Math 307. Ring theory continues in Chapter IX (Sections 45-47), which concentrates on the various classes of rings based on factorization properties. This will be followed by Chapter VI (Sections 29-33), which covers the topic of Extension Fields. We will then look at a few of the application of field extensions, which solve some of the ancient problems of the Greeks on geometric constructions. Finally, we will finish the semester by covering as much as Chapter X as time allows, which covers the topic of Galois Theory. Galois Theory will link all of the topics covered previously in the following way: Field extensions are contructed by polynomial rings modulo a maximal ideal, and Galois theory connects the subgroup structure of the automorphism group of a field extension (which is an instance of a group action) to the subfield structure of that field extension.

Dates & Course Announcements:

Exam Calendar (tentative):

There will be one Midterm exam, and a Final exam. Both will consist of a take-home portion and a timed portion. The set date and time of the Final is when the timed portion of the Final exam will be held. The time for the midterm is tentative.
Exam: Date: Time: Location:
Midterm Fri, Mar 15 In Class
Midterm Week of Mar 22 Take home
Final Exam Tues, Apr 30 9-12 Jones 306
  • 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, Jan 21: NO CLASS (MLK Holiday)
    • Mon, Jan 28: ADD/DROP DEADLINE
    • Sat, Mar 2 - Sun, Mar 10: NO CLASS (Spring Break)
    • Fri, Mar 15: WITHDRAW DEADLINE
    • Tues, Apr 30: FINAL EXAM
  • (1/16) I am still figuring out when my regular office hours will be for the semester, but for this first short week, I will be available in my office (Jones 130) on Wed, Jan 16, and Thurs, Jan 17 from 2:30 until 4:30.
  • (1/23) My office hours this week (still not permanent) will be again Wed (Jan 23) and Thurs (Jan 24), 2:30-4:30.
  • (1/24) Here is a pdf of some notes on Group Actions. There is some material in these notes which is not in Section 16 of Fraleigh, and likewise, some material in Section 16 of Fraleigh which is not in these notes.
  • (2/8) There is no problem set due on Fri, Feb 15, but you have a reading assignment: read Sections 18 and 19 in Fraleigh in detail. These sections cover the basic notions regarding rings, and all of this material was covered in Math 307 (with the exception of the notion of a skew field). I will expect you to know all of this material when I begin talking about rings next week (beginning with a review of ring homomorphisms and ideals in Section 26, which you could also read).
  • (2/8) Here are some notes on commutator subgroups and their relevance to solvability.
  • (3/1) Here is a link for an article about the axioms which define a Euclidean domain. You must be either on campus ethernet, or logged in remotely through Swem to view it. It is only two pages, so take the time to read it and understand it.
  • (3/21) Here are some brief notes with a proof that the nonzero elements of a finite field form a cyclic group under multiplication. The part of the notes which we did not go through in class is the proof of Lemma 1, so please read and understand the proof of it.
  • (4/8) I fixed a small, but significant, typo in the handout problem for HW #7. In part (c), the assumption should be c ≥ 5 (not c=5 as it incorrectly read before).
  • (4/22) My office hours today, Mon, Apr 22, will be 4-5 instead of 1:30-2:30. I will have the following office hours for you all leading up to the final exam: Wed Apr 24 2:30-4:30, Thurs Apr 25 9:30-10:30 and 3:30-5, Fri Apr 26 3:30-5, and Mon Apr 28 12-2 and 3:30-5.


Problem sets for homework will be assigned roughly every week, or sometimes every other week. For each homework problem assigned, a complete solution explaining each step should be written up (hand-written, or using LaTex). Be sure to explain your steps and reasoning for calculations as well as for proofs. Homework is due at the beginning of class on the due date of the assignment. The late policy for homework is as follows, and there are no exceptions (you can always email me your homework for it to be on time if you must be out of town): Turned in on the same day, but after class: -10%, turned in by 5 pm on the next weekday: -20%, and then -20% for each day after. Your final homework grade (30% of total) will be obtained by taking your six highest HW scores. You may work with other students on problems, but each student must turn in their own write-up of their assignment. If any problems are completed in a collaborative effort, please indicate so on the paper turned in. You may not use any source on the internet for solutions.
Assignment Problems Due Date
1 pgs. 310-311 (Sec. 34) #3, 7, 8
Also, the 3 problems on this handout
Fri, Feb 1
2 Sec. 16 #9, 12, 15, Sec. 17 #6, 8
Also, the problem on this handout
Fri, Feb 8
3 Sec. 36 #16, 19, Sec. 37 #4, 5, 6, 7, 8
You may use these notes for Sec. 37 #8.
Fri, Feb 22
4 Sec. 26 #22, Sec. 27 #24, 34, 35, Sec. 22 #24, 25 Fri, Mar 1
5 Sec. 46 #12, 15, Sec. 47 #16, 18
Also, the two problems on this handout
Fri, Mar 29
6 Sec. 23 #18, 19, 21, 37, Sec. 29 #6, 7, 8
Also, the problem on this handout
For Sec. 23 #18, 19, 21, and Sec. 29 #6-8, *prove*
the irreducibility of the polynomials over Q.
Fri, Apr 5
7 Sec. 29 #29, 30, 36, 37, Sec. 30 #21, 24
Also, the problem on this handout
Fri, Apr 12
8 Sec. 31 #23, 24, 28, 29, 30, Sec. 50 #24
Optional: The problems on this handout
Fri, Apr 19

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. If you decide to do this, you must write your paper on a topic in Abstract Algebra approved by me, and you must keep to a schedule of turning in drafts that is set at the beginning of the semester in order to get credit. You are also encouraged to sign up for Math 300 during this semester if you fulfill the writing requirement through this class.


There are several opportunities for undergraduates through the William & Mary mathematics department, including research in mathematics. If you are interested, feel free to ask me or someone else in the Mathematics department about these opportunities. General information is available here.