Math 430 Abstract Algebra II,
Spring 2015
Instructor: Chi-Kwong Li
Meeting time and place: TT 11:00 - 12:20 p.m. Jones 113
Office: Jones 128, Tel: 221-2042
E-mail: ckli@math.wm.edu, http://cklixx.people.wm.edu
Office hours: TWR 9:00-10:00 a.m. or by
appointments.
Course description:
The goal of the course is to study advanced topics in Abstract Algebra and their Applications.
The course will cover Chapter 16 - 33 of the following required textbook,
and possibly supplementary material
from other reference books.
-
Joseph A. Gallian, Comtemporary Abstract Algebra, 8th edition, Brooks/Cole.
There will be weekly assignment due on Thursday 5:00 p.m. starting January 29.
Homework sessions will be conducted on wednesday, 3:00-4:00 p.m. Jones 112, 113, or 131.
You are required to use LaTex to typset the solution.
You may get the programs for free.
- For windows, you may
(a) download the MikTex program from http://miktex.org/download;
(b) then download the Texmaker program from http://www.xm1math.net/texmaker/download.html.
-
For Mac users, you may download MacTex from http://tug.org/mactex/.
- You may also use the online editor
Write LaTeX .
- Here is
a list of TeX commands for mathematics symbols.
Challenging problems will be assigned from time to time;
extra-credits will be given to successful (or partially successful)
attempts.
Assessment
Examination 1. Feb. 19, 11:00 - 12:20 p.m.
Examination 2. March 26 11:00 - 12:20 p.m.
Final Exam May 5 2:00 - 5:00 p.m.
Grades (for homework, exams, final grade, etc.):
%: 0 - 60 - 65 - 70 - 75 - 80 - 83 - 87 - 90 - 93 - 100
F D C- C C+ B- B B+ A- A
Assessment: Homework Exam. 1 Exam. 2 Final
20% 25% 25% 30%
(Extra credit problems may add another 5%)
Class notes and additional references
Homework List.
- Homework 1 (Due: January 29.)
Chapter 16. # 3, 4, 6, 13, 15, 18, 20, 24, 60. Optional: 34.
[
Sample Solution. ]
- Homework 2 (Due February 5.)
Chapter 17. # 2, 8, 12, 16, 26, 28, 30, 32, 40.
Optional. #18. You may see #17 for hints.
[
Sample Solution. ]
Hints:
#8. May assume that Z_p[x]/'<'f(x)'>' is a commutative ring. Just show that
every nonzero element has an inverse.,
# 28 Do k = 2 first, and use induction.
For k = 2, show that assume that p(x) does not divide f_1(x), then the ideal generated by
p(x), f(x) contains 1.
So, there is r(x), s(x) such that 1 = f_1(x) r(x) + p(x) s(x).
So, f_2(x) = (f_1(x)r(x) + p(x) s(x)) f_2(x), and ......
- Homework 3 (Due February 12.)
Chapter 18. # 10, 12, 16, 18, 20, 26, 28, 34.
[Hint: Consider $Z[2i]$ in $Z[i]$ for # 16.]
Optional: Prove that {\sqrt{p}: p = 1, or p is a prime} is a linearly independent set over Q.
[
Sample Solution. ]
- Exam 1. (Take home part) (Due February 19)
Chapter 20 # 8, 9, 10, 12, 18.
In class part will have 5 questions chosen from Homework 1 - 3.
[
Sample Solution. ]
- Homework 4 (Due February 26)
Chapter 20. # 2, 4, 6, 14, 20, 22, 32, 38, 40.
Optional: 28.
[
Sample Solution. ]
- Homework 5 (Due March 5)
Chapter 21. # 3, 6, 8, 16, 18, 20, 24, 26, 32, 38.
[Hint: For 32, 38, consider the dimension
of the extension fields over the ground fields.]
[
Sample Solution. ]
- Homework 6 (Due March 19)
Chapter 22. # 2, 6, 10, 14, 20, 32, 36.
Chapter 23. # 4, 6, 10 [If 40 degree is constructible, then 20 degree ....].
Optional: Chapter 22. # 26, 40.
[
Sample Solution. ]
- Homework 7 (Due March 26)
Chapter 24. #5, 6, 8, 18, 20, 26. Problems in the last page of Notes on Chapter 24.
[
Sample Solution. ]
- Exam 2 - Take Home (Due March 31)
p.397, #42 [Hint: See p. 382 for the definition of prmitive element; see the solution of #41
for idea
of proof.]
p.405, #12 [Hint: For a nonzero element a in R, try to write a^{-1} as a polynomial of a by
studying the minimal polynomial of a.]
Chapter 24. #50, 52, 60.
[
Sample Solution. ]
- Homework 8 [Due: April 9]
Chapter 25. # 8, 14, 16, 20, 24, 26, 28. Chapter 26. # 8, 24.
[
Sample Solution. ]
- Homework 9 [Due: April 16]
Chapter 29. # 2, 4, 12. Chapter 31. # 6, 10, 14, 16, 20, 24, 34.
[
Sample Solution. ]
- Homework 10 [Due: April 23]
Chapter 32. # 6, 8, 10, 12, 14, 18, 22, 24, 34.
[
Sample Solution. ]
- Final Examination. (Take Home component, Due: May 5, 2015].
Chapter 33 # 16, 18, 20. [10 points each.]
[10 point.]
Extra credits Prove or disprove that every finite group can be realized as
the Galois group Gal(F/Q) for some splitting field of a polynomial in Q[x].
There will be 6 questions [5 points each]
chosen from homeworks for the in class exam. component.
Possible writing projects for Math 300
- Determine the number of monic irreducible polynomials of degree k in Z_p[x].
- Survey the applications of abstract algebra in geometry. Say, trisection of the right angle.
- Survey the applications of abstract algebra in biology.
- Survey the applications of abstract algebra in chemistry.
- Survey the applications of abstract algebra in physics.
- Survey the applications of abstract algebra in economics.
- Survey the applications of abstract algebra in cryptology.
- Survey the applications of abstract algebra in engineering.