Tentative Syllabus for Applied Abstract Algebra, Fall 2001

1. Aug 20	m-ary notation, Nim
2. Aug 22	Euclidean algorithm + gcd
	extended version
Aug 24	math lab C119
3. Aug 27	Primes, UFT, primality testing
4. Aug 29	Congruences
	Applications (Doomsday algorithm for days of the week)
	repeated squaring
	Euler's thrm, CRT
Aug 31	math lab C119
5. Sep 3	Repeated squaring algorithm
	Euler's thrm
6. Sep 5	LFSR's and ciphers
Sep 7	math lab C119
7. Sep 10	CRT
8. Sep 12	RSA cryptosystem
Sep 14	math lab C119
9. Sep 17	Diffie-Hellman
10. Sep 19	Finite fields
Sep 21	math lab C119
11. Sep 24	F[x], factoring
	irreducible polynomials
12. Sep 26	More finite fields
Sep 28	math lab C119
13. Oct 1	Error correcting codes
	basic definitions
14. Oct 3	Hamming codes
Oct 5	math lab C119
15. Oct 8	Decoding Hamming codes
16. Oct 10	Cyclic codes
Oct 12	math lab C119

11. Oct 15	Permutations
12. Oct 17	Cycle notation
Oct 19	math lab C119
13. Oct 22	Symmetric groups
	permutation groups
14. Oct 24	Examples
	symmetry gps
	Rubik's cube gp
	monomial matrix gp
Oct 26	math lab C119
15. Oct 29	BCH codes
16. Oct 31	decoding BCH codes
Nov 2	math lab C119

Two papers due: one biography, one math paper (with at least one proof).

Computer projects: assigned MAPLE, MAGMA projects.

MAGMA coding theory exercises

MAPLE coding theory exercises

MAPLE group theory exercises

Homework: assigned.

Exercises so far:
1. Exercise 1.3.10, page 19,
2. Exercise 1.4.2, page 20-21,
3. Exercise 1.4.8, page 22
4. Exercise 1.4.12, page 25
5. Exercise 1.7.23, page 64 (use MAPLE or GAP or MAGMA)
6. Exercise 1.7.19, page 62
7. Exercise 1.7.26, page 66.
8. Define the sequence {s_n}_n=1,2,... in Z/19Z by s_i+2=s_i+1+s_i, s₁=s₂=1. Using the roots r₁ and r₂ of x²-x-1=0 in Z/19Z, solve for s_n in the form s_n=c₁r₁ⁿ+ c₂r₂ⁿ, for n=1,2,... Check your work by computing s₃, s₄, s₅ in two ways (one, using the definition, the other using your formula). What is the period of this sequence? (You may use GAP or MAGMA or MAPLE, if you like.)
9. Exercises 1-5 in the "Coding theory exercises in GAP" handout.
10. Exercise 3.2.24, page 147.
11. Exercise 5.3.3, page 199.
12. Exercise 5.3.17, page 203.
13. Exercise 5.3.19, page 203.
14. Exercise 5.6.5, page 213.
15. Exercise 5.7.3, page 214.
16. Exercise 5.7.5, page 215.
17. Exercise 5.8.6, page 219.
18. Exercise 5.9.4, page 221.

Rough draft of project due: Nov 26
Presentations: Nov 30th and Dec 3rd
Project due: Dec 10

Tests and quizzes: as announced.

Text: Applied abstract algebra notes by D. Joyner, R. Kreminski, J. Turisco

Talks of interest:

1. Oct 17, Don Newhart, NSA
2. Dec 5, Gene Berg, NSA

Policy statement

Downloading GAP 4.3 (including share packages and all documentation) for windows: go to the web page installing gap 4.3 for windows