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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.