MA257 Content
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