# 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.

**Exercises so far**:- Exercise 1.3.10, page 19,
- Exercise 1.4.2, page 20-21,
- Exercise 1.4.8, page 22
- Exercise 1.4.12, page 25
- Exercise 1.7.23, page 64 (use MAPLE or GAP)
- Exercise 1.7.19, page 62
- Exercise 1.7.26, page 66.
- Define the sequence {sn}n=1,2,... in
**Z**/19**Z**bysi+2=si+1+si, s1=s2=1. Using the roots r1 and r2 of x2-x-1=0 in**Z**/19**Z**, 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.) - Exercises 1-5 in the "Coding theory exercises in GAP" handout.
- Exercise 3.2.24, page 147.
- Exercise 5.3.3, page 199.
- Exercise 5.3.17, page 203.
- Exercise 5.3.19, page 203.
- Exercise 5.6.5, page 213.
- Exercise 5.7.3, page 214.
- Exercise 5.7.5, page 215.
- Exercise 5.8.6, page 219.
- 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: