Math 372: Introduction to Number Theory
In Fall 2005 I taught a course in number theory. Much of the course material was from
my own handouts.
Handouts
Background Assumptions for Number Theory
The Quotient-Remainder Theorem
sometimes called the Division Algorithm
Fundamental Theorem of Arithmetic
sometimes called the Unique Factorization Theorem
Irrationality of Square Roots
Infinite Number of Primes
(first proved by Euclid)
Congruence and Modulus: Part 1
Congruence and Modulus: Part 2
Chinese Remainder Theorem
Euler's Phi and Euler's Theorem
Fields and Polynomials
(including a theorem of Lagrange)
Cyclic Unit Groups
(including the Primitive Element Theorem)
Quadratic Residues
Other Topics
Hasse Principle
Crypto article in
Wired magazine.
Course information
Prerequisites:
Experience with mathematical proofs including
proofs that use mathematical induction.
This experience can be obtained by taking Math 350 or Math 370.
Textbook
Elementary Number Theory
by Gareth A. Jones, Josephine M. Jones
ISBN: 3540761977
Paperback: 200 pages
Publisher: Springer (1998)
The primary source for this class will be the lectures,
but this textbook will serve as an important reference, and source of exercises.
Schedule
UNIT 1. Divisibility, Factorization, Congruence. This unit begins with basic facts about divisibility and builds to the Unique Factorization Theorem (Fundamental Theorem of Arithmetic). Along the way we will study Bezout's identity, linear Diophantine equations, and prime numbers. At the end of this unit, we will study the definition and basic properties of congruences. This unit corresponds roughly to Sections 1.1 to 3.1 in the textbook.
UNIT 2. Structure of the mod n ring. This unit focuses on the structure of the ring of integers modulo n. The case where n is a prime number will be emphasized. Topics include linear congruences, the Chinese remainder theorem, Fermat's little theorem, Euler's theorem, Euler's phi function, Wilson's theorem, pseudoprimes and Carmichael numbers, and units. The notions of group, ring, and
field will be introduced. This unit corresponds to most, but not all, of Chapters 3 - 6 in the textbook.
UNIT 3. Further topics. This unit will focus on quadratic reciprocity, and a selection of other topics not covered in Units 1 and 2. This corresponds to a selection from Chapters 6 to 10 in the textbook.
UNIT 4. Diophantine equations and the Local-Global Principle. This unit will focus on an important topic in modern number theory: the Local-Global Principle. This unit will be based on materials supplied by me including a paper written by Franz Lemmermeyer and myself.
Instructor Information
Dr. Wayne Aitken
Office: SCI2 327
Telephone: (760) 750-4155 (but I prefer e-mail)
e-mail:
waitken@csusm.edu