Math 422: Introduction to Number Theory
Website for Math 422 (Spring 2010).
Last updated on 19 January 2010.
Course information
This is a introductory course in number theory
designed for junior and senior level mathematics
majors.
Prerequisites:
Math 378: Number Systems. Students need to know basic facts
about primes, divisors, GCDs, prime factorization, and
modular arithmetic. Students also need to know some
algebraic terminology including
the definitions of group, ring,
field, and integral domain.
Textbook Policy
Most of the course will be based on the lecture notes included in this website.
In addition, there one required textbook. This textbook will be used as a source of exercises,
and as an additional reference. In addition, we may cover some sections of this book in class.
Elementary Number Theory
Gareth A. Jones, Josephine M. Jones
Springer (corrected edition 1998)
Lecture Notes
Lecture Notes 1.
Bezout's identity, Euclidean algorithm.
Lecture Notes 2.
Unique factorization.
Lecture Notes 3.
Applications of the Order function.
Lecture Notes 4.
Prime numbers.
Lecture Notes 5.
Linear Equations.
Lecture Notes 6.
Chinese Remainder Theorem.
Lecture Notes 7.
Fermat's Little Theorem, Euler's Theorem,
Wilson's Theorem.
Lecture Notes 8. Polynomials.
Lecture Notes 9.
Primitive Roots.
Lecture Notes 10.
Quadratic Residues.
Lecture Notes 11.
"Counterexamples to the Hasse Principle: An Elementary
Introduction" (with F. Lemmermeyer)
Lecture Notes 12.
RSA cryptosystem.
Background
The background for this course is covered in our Math 378 course (number
systems). Here are summaries for the first half of Math 378 covering
most of what you need to know. We will spend a few weeks in class reviewing
this material.
Summary of Chapter 1
Natural Numbers N.
Summary of Chapter 2
Background materials from Math 378 (Subtraction in N, Well-ordering,
counting, finite and infinite sets).
Summary of Chapter 3
Background materials from Math 378 (The ring Z)
Summary of Chapter 4
Background materials from Math 378 (Absolute values, induction variants, divisibility, quotient-remainder, GCD, LCM, primes, summation notation, finite product notation, factorization into primes, infinitude of primes, base B expansion)
Summary of Chapter 5
Congruences, modular arithmetic, fields, the field with p elements.
Summary of Chapter 6
Rational numbers as an ordered field.
Instructor
Prof. Wayne Aitken
e-mail:
waitken@csusm.edu