MA257 Introduction to Number Theory
Lecturer: Dr. Adam Harper
Term(s): Term 2
Status for Mathematics students: List A
Commitment: 30 one hour lectures
Assessment: 2 hour Exam 85%, Homework Assignments 15%
Prerequisites: MA136 Introduction to Abstract Algebra
Co-requisite: MA249 Algebra II: Groups and Rings
Leads To: MA3A6 Algebraic Number Theory, MA426 Elliptic Curves
Content:
• Factorisation, divisibility, Euclidean Algorithm, Chinese Remainder Theorem.
• Congruences. Structure on Z/mZ and its multiplicative group. Theorems of Fermat and Euler. Primitive roots.
• Quadratic reciprocity, Diophantine equations.
• Elementary factorization algorithms.
• Introduction to Cryptography.
• p-adic numbers, Hasse Principle.
• Geometry of numbers, sum of two and four squares.
Aims:
To introduce students to elementary number theory and provide a firm foundation for later number theory and algebra modules.
Objectives:
By the end of the module the student should be able to:
- work with prime factorisations of integers
- solve congruence conditions on integers
- determine whether an integer is a quadratic residue modulo another integer
- apply p-adic and geometry of numbers methods to solve some Diophantine equations
- follow advanced courses on number theory in the third and fourth year
Books:
H. Davenport, The Higher Arithmetic, Cambridge University Press.
G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, Oxford University Press, 1979.
K. Ireland and M. Rosen, A Classical Introduction to Modern Number Theory, Springer-Verlag, 1990.
Additional Resources
Archived Pages: 2015