Lecturer: Hong Liu
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
- MA136 Introduction to Abstract Algebra: Rings, subrings, ideals, integral domains, fields.
- MA132 Foundations or MA138 Sets and Numbers: Congruence modulo n, prime factorisation, Euclidean algorithm, gcd and lcm, Bezout Lemma.
Useful background: Interest in Number Theory is essential.
- MA249 Algebra II: Groups and Rings: Ring theoretic part Algebra II and Introduction to Number Theory have much in common.
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
- 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
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.