McGill University

Department of Mathematics & Statistics

Basic Algebra I

189-235A

Detailed Syllabus

1. Sept. 2-Sept 6: (Appendix A-D). Overview of the course. Sets, relations, and functions. Induction. Number systems including complex numbers. Equivalence relations.
   
2. Sept 9-Sept 13: (Chapter 1). Arithmetic in Z. The division algorithm. The Euclidean algorithm and gcd's. Fundamental theorem of arithmetic. Prime numbers.
   
3. Sept 16-Sept 20: (Chapter 2). Congruences. Modular arithmetic. Finite fields. Primality testing.
   
4. Sept 23-Sept 27: (Chapter 3). Rings. Definitions and basic examples. Isomorphisms and homomorphisms.
   
5. Sept 30-Oct 4: (Chapter 4). Arithmetic in polynomial rings. Division algorithm and unique factorization.
   
6. Oct 7-Oct 11: (Chapter 4, cont'd).
   
7. Oct 14-Oct 18: Review of the material, and midterm test.
   
8. Oct 21-Oct 25: (Chapter 5). Congruences in polynomial rings. More on finite fields.
   
9. Oct 28-Nov 1: (Chapter 6). Ideals and quotient rings.
   
10. Nov 4-Nov 8: (Chapter 9). Integral Domains. Unique factorization in number rings. Application to quadratic integers.
    
11. Nov 11-Nov 15: (Chapter 9). More on rings, Integral domains, and quadratic integers.
    
12. Nov 18-Nov 22: (Section 7.1-7.5). Group theory. Definition and basic examples. Subgroups, isomorphism and homomorphism. Lagrange's theorem.
    
13. Nov 25-Nov 29: (Section 7.6-7.9). More group theory. Normal subgroups, quotients, and homomorphisms. Simple groups.
14. Dec 2-Dec 4: Review of the material.