McGill University

Department of Mathematics & Statistics

Basic Algebra I

189-235A

Detailed Syllabus

1. Sept. 1-Sept 3: Overview of the course. Sets, relations, and functions. Induction. Number systems including complex numbers. Equivalence relations.
   
2. Sept 8-Sept 10: (Chapter 1) Arithmetic in Z. The division algorithm. The Euclidean algorithm and gcd's.
   
3. Sept 13-Sept 17: (Chapter 1) Fundamental theorem of arithmetic. Prime numbers.
   
4. Sept 20-Sept 24: (Chapter 2) Congruences. Modular arithmetic. Finite fields. Primality testing.
   
5. Sept 27-Oct 1: (Chapter 3) Rings. Definitions and basic examples. Isomorphisms and homomorphisms.
   
6. Oct 4-Oct 8: (Chapter 4) Arithmetic in polynomial rings. Division algorithm and unique factorization.
   
7. Oct 11-Oct 15: (Chapter 5) Congruences in polynomial rings. More on finite fields.
   
8. Oct 18-Oct 22: Review of the material, and midterm test (to be given, in-class, on Friday October 22.)
   
9. Oct 25-Oct 29: (Chapter 6) Ideals and quotient rings.
   
10. Nov 1-Nov 5: (Chapter 9) Integral Domains. Unique factorization in number rings. Application to quadratic integers.
    
11. Nov 8-Nov 12: (Chapter 9) More on rings, Integral domains, and quadratic integers.
    
12. Nov 15-Nov 19: (Sections 7.1 -7.5) Group theory. Definition and basic examples. Subgroups, isomorphism and homomorphism. Lagrange's theorem.
    
13. Nov 22-Nov 26: (Sections 7.6-7.9) More group theory. Normal subgroups, quotients, and homomorphisms. Simple groups.
14. Nov 29-Dec 3: Review of the material.