## McGill University

# Department of Mathematics & Statistics

# Basic Algebra I

# 189-235A

## Detailed Syllabus

** Sept. 2-Sept 9**:
Overview of the course. Sets,
relations, and functions.
Induction. Number systems
including complex numbers. Equivalence relations.

** Sept 12-Sept 16**: (Chapter 1)
Arithmetic in **Z**. The division algorithm.
The Euclidean algorithm and gcd's.

** Sept 19-Sept 23**: (Chapter 1)
Fundamental theorem of arithmetic.
Prime numbers.

** Sept 26-Sept 30**: (Chapter 2)
Congruences. Modular arithmetic. Finite fields.
Primality testing.

** Oct 3-Oct 7**: (Chapter 3)
Rings. Definitions and basic examples. Isomorphisms and homomorphisms.

** Oct 10-Oct 14**: (Chapter 4)
Arithmetic in polynomial rings. Division algorithm and
unique factorization.

** Oct 17-Oct 21**: (Chapter 5)
Congruences in polynomial rings. More on finite fields.

** Oct 24-Oct 28**:
Review of the material, and midterm test (to be
given, in-class, on Friday October 28.)

** Oct 31-Nov 4**: (Chapter 6)
Ideals and quotient rings.

** Nov 7-Nov 11**: (Chapter 6)
Ideals and quotient rings. Introduction to groups.

** Nov 14-Nov 18**:
(Sections 7.1 -7.5)
Group theory. Definition and basic examples. Subgroups,
isomorphism and homomorphism. Lagrange's theorem.

** Nov 21-Nov 25**:
(Sections 7.6-7.9)
More group theory. Normal subgroups, quotients, and homomorphisms.
Simple groups.

** Nov 28-Dec 2**:
Review of the material.
** Dec 5**: Final exam.