Math 5286: Honors fundamental structures of algebra II

Spring 2000

Prerequisites: Math 5285 or the equivalent, that is, a solid, rigorous background in theoretical linear algebra and group theory.  
Instructor: Victor Reiner (You can call me "Vic"). 
Office: Vincent Hall 256, Telephone (with voice mail): 625-6682, E-mail: 
Classes: Mon-Wed-Fri 09:05-09:55am in Vincent Hall 209.  
Office hours: Monday and Wednesdays 11:15-12:05;
one can always see me by appointment.  
Course content: This is the second semester of a theoretical course in the basic algebra of groups, ring, fields, and vector spaces. In Math 5285 in the Fall, we covered vector spaces, linear algebra, and some group theory, completing most of Chapters 1-5 in the text by Artin indicated below.
Required text: Algebra, by Michael Artin, Prentice-Hall, 1991.
As far as what we will cover in the book this semester, we intend to start with a very brief detour into Chapter 7 to do a version of the Spectral Theorem. Then we'll continue group theory by doing Chapter 6. Then we'll skip the rest of Chapter 7, and all of Chapters 8, 9, to plow onward into ring, module, and field theory by doing as much of Chapters 10-14 that we can manage. We will certainly skip the later parts of Chapter 11 (Sections 11.8 through 11.12), of Chapter 13 (Sections 13.7, 13.8, 13.9), and of Chapter 14 (Sections 14.6 through 14.9).
Other useful texts: Topics in algebra, by I. N. Herstein, Xerox College Publishing, 1975.
Contemporary abstract algebra, by Joseph Gallian,Houghton Mifflin, 1998.
Homework: As in the Fall, there will be homework assignments due every two weeks in class on Fridays during the semester, starting with Friday September 17 (see table below). There should be a total of 6-7 assignments, which will count for 40% of the course grade. The assignments will mainly be exercises from the book. Late homework will not be accepted. I encourage collaboration on the homework, as long as each person understands the solutions, writes them up in their own words, and indicates on the homework page with whom they have collaborated.  
Exams:  There will be two take-home midquarter exams to be handed out on dates to be announced later, each contributing 15% to the grade. There will be a take-home final exam worth 30% of the grade given during exam period. In contrast to the homework, there is to be no collaboration allowed with other humans allowed on any of the take-home midquarter or final exams. 
Homework assignments
Assignment Due date Problems
1 1/28 5.6 # 2,3
5.7 # 2
5.9 # 3,4,5
Chap. 5 Misc. Probs. # 7,8,9
7.5 # 5
2 2/11 6.1 # 1,4,5,6,14
6.2 # 4,6,8,10,2 (Do #10 before doing #2 !)
6.3 # 3,7,9,13
3 3/3 6.6 #3,4,5,15,16,20
10.1 #2,4,8,9,11,12,13
4 3/17 10.3 #3,4,7,8,9,17,24,26,30,33
10.4 #3,6
5 4/14 10.6 # 4,5
10.7 # 1,2,7,10
13.1 # 1
13.2 # 1,5
13.3 # 1,2,3,8,10,12,15
6 4/28 13.4 # 5,6,7
13.5 # 1,2,3
13.6 # 2,3,4,5,9,10,11
Midterm exams
Exam Due date Problems
Midterm 1 Due 2/18 6.1 # 7, 12
6.3 # 12
6.4 # 1,2,6,15
Midterm 2 Due 3/27 10.5 #2,3,5,7,9,14,15
10.6 #2
Extra Problems:
1.(a) Prove the following
For F a field, the units of F[x] are the units of F, i.e. the elements F-{0}.
For any ring R, the units of R[x] contain the units of R, and are contained
in the set of polynomials whose constant term is a unit of R.

(b) Prove that both inclusions in the second assertion of the proposition can be strict,
by exhibiting two polynomials in Z/4Z[x] of the form 1+ax with a not 0,
one of which is a unit in Z/4Z[x], and one which is not.
Final Exam Due 5/5 11.1 #1,4
11.2 #2,4,8,10
11.3 #4
Chap. 13 Misc. Probs. #2
Extra Problems:
1. Let p be a prime, and n,m positive integers with p-ary expansions
n = a0 + a1 p + a2 p^2 + ... + ar p^r
m = b0 = b1 p + b2 p^2 + ... + br p^r.

(a) Show that the binomial coefficient (n choose m) is congruent mod p to
the product (a0 choose b0) (a1 choose b1) ... (ar choose br).
(Hint: consider the coefficient of x^m in
the expansion of (x+1)^n when working in Z/pZ[x] ).

(b) Use (a) to determine when (n choose m) is odd.
Describe exactly those values of n which have the property
that (n choose m) is odd for all m less than or equal to n.

(c) Show (pn choose pm) is congruent mod p to (n choose m).
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