MATH 4034 - Number Theory

Description

This course introduces the fundamental concepts and results of elementary number theory. Topics include divisibility, primes, the Euclidean algorithm, the Fundamental Theorem of Arithmetic, congruences, linear and polynomial congruence equations, the Chinese Remainder Theorem, multiplicative functions, primitive roots, quadratic residues, the Law of Quadratic Reciprocity, and selected classical results. Proof-writing and problem-solving are integral components of the course.

Learning Objectives

• Apply the Euclidean algorithm and the Fundamental Theorem of Arithmetic.
• Analyze multiplicative functions, including the Euler phi-function.
• Solve congruence equations, including systems of congruences, using the Chinese Remainder Theorem and other methods.
• Analyze number-theoretic problems using Wilson’s, Fermat’s, and Euler’s theorems.
• Use quadratic residues, including applying the Law of Quadratic Reciprocity.
• Prove results using a variety of techniques.