## Mathematics 3101 Algebraic Structures with Computer Applications

 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.