David Joyner, Math Department

# Applied Abstract Algebra

Tentative Syllabus for Applied Abstract Algebra, SM486 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, GAP projects.

Homework: assigned.

1. Exercises so far:
2. Exercise 1.3.10, page 19,
3. Exercise 1.4.2, page 20-21,
4. Exercise 1.4.8, page 22
5. Exercise 1.4.12, page 25
6. Exercise 1.7.23, page 64 (use MAPLE or GAP)
7. Exercise 1.7.19, page 62
8. Exercise 1.7.26, page 66.
9. Define the sequence {sn}n=1,2,... in Z/19Z by si+2=si+1+si, s1=s2=1. Using the roots r1 and r2 of x2-x-1=0 in Z/19Z, solve for sn in the formsn=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.)
10. Exercises 1-5 in the "Coding theory exercises in GAP" handout.
11. Exercise 3.2.24, page 147.
12. Exercise 5.3.3, page 199.
13. Exercise 5.3.17, page 203.
14. Exercise 5.3.19, page 203.
15. Exercise 5.6.5, page 213.
16. Exercise 5.7.3, page 214.
17. Exercise 5.7.5, page 215.
18. Exercise 5.8.6, page 219.
19. 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 by D. Joyner, R. Kreminski, J. Turisco

Talks of interest:

Policy statement