General Information:
| Meeting Time: | MWF, 11:00 - 11:50 |
| Location: | Jones 306
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| 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.
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| Textbook: | A First Course in Abstract Algebra, Seventh Edition, by
John B. Fraleigh |
Grade
Breakdown: | 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.
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| Prerequisite: | Math 307 - Abstract Algebra I. |
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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 |
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| Midterm |
Week of Mar 22 |
Take home |
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| Final Exam |
Tues, Apr 30 |
9-12 |
Jones 306
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- 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.
Homework:
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
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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.
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Fri, Apr 5 |
| 7 | Sec. 29 #29, 30, 36, 37, Sec. 30 #21, 24
Also, the problem
on this handout
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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.
Research:
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.
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