MA257 Introduction to Number Theory
Lecturer: Sam Chow
Term(s): Term 2
Status for Mathematics students: List A
Commitment: 30 one hour lectures
Assessment: 85% 2 hour examination, 15% homework assignments
Formal registration prerequisites: None
Assumed knowledge:
MA267 Groups and Rings or MA268 Algebra 3
- Ring theory: rings, subrings, ideals, integral domains, fields
- Number theory: congruence modulo n, prime factorisation, Euclidean algorithm, gcd and lcm, Bezout Lemma
Useful background: Interest in Number Theory is essential
Leads to: The following modules have this module listed as assumed knowledge or useful background:
- MA3G6 Commutative Algebra
- MA3A6 Algebraic Number Theory
- MA4H9 Modular Forms
- MA4L6 Analytic Number Theory
- MA426 Elliptic Curves
- MA4H8 Ring Theory
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