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.
|Meeting Time:||MWF, 11:00 - 11:50
|Location: ||Jones 307|
Office: Jones 130
Office Hours: Mon 1-2, Tues 9-10, and Thurs 3-4:30, or by appointment/walk-in.
|Textbook:||A First Course in Abstract Algebra, Seventh Edition, by
John B. Fraleigh|
| Grade |
| 1 Midterm Exam - 30%, Homework
- 30%, Final Exam - 40%. The
grading scale will roughly be a 10 percentage point scale, so that a final
score of 90% is in the A range, a score of 80% is in the B range, etc. |
|Prerequisite: || Math 307 - Abstract Algebra I. ||
Dates & Course
Exam Calendar (tentative):
||Mon, May 9
- All relevant announcements will be listed here. Check back frequently (don't forget
to refresh your browser) for updates.
- Important Dates and Class Holidays:
- Sat, Mar 5 - Sun, Mar 15:
NO CLASS (Spring Break)
- Mon, May 9: FINAL EXAM
- 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. 1/19 and Thurs. 1/20 from 3:00 until 4:45.
- 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.
- (1/31) My office hours on Tues, Feb 1, will be 10:30-11:30 instead of
9-10. All other office hours this week will remain the same as usual.
- Here are some notes on the conjugacy classes
of symmetric groups. Use them as a guide if you get stuck on Problem 8 in
Section 37 of the homework.
- Here are some notes on the commutator
subgroup of a group. The notes contain the proofs of the statements which
are applied in Example 37.15 in the text. These methods can be applied to
solve Problem 4 in Section 37 on the homework.
- (2/14) My office hours today, Mon Feb 14, will be 1:30-2:30 instead of
- (2/25) 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.
- (2/25) The midterm will have an in-class part, which will be on Fri Mar 1,
and a take-home part, which will be given to you on Mon Mar 14, and due on
Mon Mar 21. The in-class portion will cover material through Wed Feb 23,
and the take-home part will cover material up to the day you get it.
- (3/2) I will not have office hours tomorrow, Thurs Mar 3, but I will have
office hours today, Wed Mar 2, 2:30-3:30. Please come if you have any
questions before the in-class midterm on Fri Mar 4.
- (4/25) EXTRA OFFICE HOURS: I will have extra office hours for help on the
last homework today, Mon Apr 25, 4-5 PM, and tomorrow, Tues Apr 26, 2:30-4
Homework and Exams:
There will be one mid-term exam and the final exam. Both will have take-home
elements, as well as in-class portions. The in-class portions will be
significantly shorter than the take-home parts, concentrating on definitions,
main concepts, and quick problems. The take-home portions of the exams will
consist of longer problems, resembling the more involved problems assigned as homework.
Problem sets for homework will be assigned roughly 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, and there are heavy penalties for
late homework (just make it easy on both of us and turn things in on time). Your final
homework grade (30% of total) will be obtained by taking your five 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.
|1 ||Sec. 34 #3, 5, 7, 8
Sec. 16 #9, 11
|Fri, Jan 28
|2 ||Sec. 36 #16, 19, 20
Sec. 37 #4, 5, 6, 7, 8
|Fri, Feb 11
|3 ||Sec. 22 #24, 25, Sec. 23 #34, 35, Sec. 26 #22
If R is
a ring with no zero divisors, f(x) and g(x) are nonzero
R[x], then show deg(f(x)g(x)) = deg(f(x)) + deg(g(x)).
|Fri, Feb 25
|4 ||Sec. 46 #12, 15, 16
Sec. 47 #15, 16, 18
This problem: pdf.
|Fri, Apr 1
|5 ||Sec. 29 #29, 30, 31
Sec. 31 #24, 28, 29, 30
|Fri, Apr 15
|6 ||Sec. 32 #5, Sec. 50 #18, 24
Sec. 33 #10, 11, 12, 13
|Tues, Apr 26
5 PM, my office
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.