Applied Abstract Algebra  

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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, MAGMA projects.

MAGMA coding theory exercises MAPLE coding theory exercises MAPLE group theory exercises

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 or MAGMA)
  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 form
    sn=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 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

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