## McGill University

# Department of Mathematics & Statistics

# Basic Algebra I

# 189-235A

## Detailed Syllabus

** Sept. 2-Sept 6**: (Appendix A-D).
Overview of the course. Sets,
relations, and functions.
Induction. Number systems
including complex numbers. Equivalence relations.

** Sept 9-Sept 13**: (Chapter 1).
Arithmetic in **Z**. The division algorithm.
The Euclidean algorithm and gcd's.
Fundamental theorem of arithmetic.
Prime numbers.

** Sept 16-Sept 20**: (Chapter 2).
Congruences. Modular arithmetic. Finite fields.
Primality testing.

** Sept 23-Sept 27**: (Chapter 3).
Rings. Definitions and basic examples. Isomorphisms and homomorphisms.

** Sept 30-Oct 4**: (Chapter 4).
Arithmetic in polynomial rings. Division algorithm and
unique factorization.

** Oct 7-Oct 11**: (Chapter 4, cont'd).

** Oct 14-Oct 18**: Review of the material, and midterm test.

** Oct 21-Oct 25**: (Chapter 5).
Congruences in polynomial rings. More on finite fields.

** Oct 28-Nov 1**: (Chapter 6).
Ideals and quotient rings.

** Nov 4-Nov 8**: (Chapter 9).
Integral Domains. Unique factorization in number rings. Application to
quadratic integers.

** Nov 11-Nov 15**: (Chapter 9).
More on rings, Integral domains, and quadratic integers.

** Nov 18-Nov 22**: (Section 7.1-7.5).
Group theory. Definition and basic examples. Subgroups,
isomorphism and homomorphism. Lagrange's theorem.

** Nov 25-Nov 29**: (Section 7.6-7.9).
More group theory. Normal subgroups, quotients, and homomorphisms.
Simple groups.
** Dec 2-Dec 4**:
Review of the material.