## Mathematics 3101

Algebraic Structures with Computer Applications

**Course Outline****Notice Board**My office hours during the exam period. Dec. 16 (Wednesday): 11:30am - 1:00am. Dec. 18 (Friday): 11:00pm -12:30pm.

Topics covered in the class.

- Sept. 10: Introduction; Sets, mappings.
- Sept. 15: Mappings, Binary operations, Binary relation.
- Sept. 17: Binary relation, divisibility, greatest common divisor
- Sept. 22: Euclidean algorithm, unique factorization, congruences.
- Sept. 24: solve linear congruence, Chinese remainder theorem, congruence classes.
- Sept. 29. Congruence classes, RSA.
- Oct. 1. Monoids.
- Oct. 6. Deterministic Finite Automata, recognizable language, regular language.
- Oct. 8. Regular language and monoids of row monomial matrices.
- Oct. 13. Groups, examples and properties.
- Oct. 15. proof of properties, subgroups, exponents and multiples.
- Oct. 20. subgroups, cyclic groups.
- Oct. 22. generators of cyclic groups, Isomorphisms.
- Oct. 27: Homomorphisms, permutation groups.
- Oct. 29: permutation groups, Cayley's theorem.
- Nov. 3: Cosets, normal subgroups.
- Nov. 5: Normal subgroups, Quotient groups.
- Nov. 10: Fundamental theorem of homomorphisms.
- Nov. 12: Second/third isomorphic theorem, rings and fields.
- NOv. 17: properties of rings, integral domains and fields, units.
- Nov. 19: ideals and quotient rings.
- Nov. 24: Maximal ideals and fields, polynomials.
- Nov. 26: division algorithms, gcd.
- Dec. 1: irreducible polynomials, constructions of finite fields.
- Dec. 3: review.
Tutorials

- Tutotial #1: Section 1.1: 19 Section 1.2 : 2, 4(a)(d)(e) Section 1.3: 4, 13 Section 2.3: 3, 7, 13, 26 Section 2.4: 3(c)(d).
- Tutotial #2: Section 2.5: 15, 23, 28, 29(l), 39, 53 (a)(e) Section 2.6: 5 (a)(b), 7(a)(b), 8(a)(b), 16.
- Tutotial #3: See WebCT.
- Tutorial #4: Section 3.3. 1,15, 17 Section 3.4: 14, 16, 18, 21, 37 Section 3.5: 7, 10, 16
- Tutorial #5: Section 4.1: 1, 3, 5, 9, 10, 11 Section 4.2: 8. Section 4.4: 3, 9, 11, 13, 21, 22. Section 4.5: 16-20. Section 4.6: 14, 17.
- Tutorial #6: Section 5.1: 21. Section 5.2: 11. Section 6.1: 4.
- Tutorial #7: Section 8.2: 1, 7, 9. Section 8.3: 4. Section 8.6: 2.
More suggested exercises:

- 1.1: 1-15.
- 1.2: 1-10.
- 1.4: 1-13.
- 1.7: 1-5, 11-12.
- 2.3: 1-3, 6-7, 17-18, 24-27.
- 2.4: 2-3, 7-14, 18-25, 29-30, 39-45, 53.
- 2.5: 1- 50, 53.
- 2.6: 1-14.
Test #1 covers the above topics

- 2.8: 17-22. (not section 2.7!)
- 3.1: 1-20: decide whether it is a monoid or not.
- Automata (tutorial #3)
- 3.1: 1-35.
- 3.2: 1-22.
Test #2 covers (section 2.7, monoids and automata theory, sections 3.1, 3.2).

- 3.3: 1-29;
- 3.4: 1-29;
- 3.5: 1-20.
- 3.6: 1-21.
- 4.1: 1-20.
- 4.2: 1-6, 8-10.
Test #3 covers up to here.

- 4.4: 3, 5, 7-13.
- 4.5: 1-7.
- 4.6: 1-9, 13-26, 28-29, 36.
- 5.1: 1-2, 7-8, 11-40.
- 5.2: 1, 11, 15, 20, 23-24.
Test #4 covers up to here.

- 6.1: 1, 4, 6, 7, 8, 24.
- 6.2: 6, 8, 9, 10.
- 6.4: 3, 4, 9, 10.
- 8.2: 1-14, 19-22.
- 8.3: 4.
**8.6: 1-6.**