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

Year 1 regs and modules
G100 G103 GL11 G1NC

Year 2 regs and modules
G100 G103 GL11 G1NC

Year 3 regs and modules
G100 G103

Year 4 regs and modules
G103

Past Exams
Core module averages