Unit information: Number Theory in 2013/14

Unit name	Number Theory
Unit code	MATH30200
Credit points	20
Level of study	H/6
Teaching block(s)	Teaching Block 2 (weeks 13 - 24)
Unit director	Professor. Wooley
Open unit status	Not open
Pre-requisites	Level 1 Pure Mathematics
Co-requisites	None
School/department	School of Mathematics
Faculty	Faculty of Science

Description including Unit Aims

Number theory concerns itself with the properties of the integers and rational numbers, although aspects of real and complex numbers, algebraic curves, matrices etc are also considered. The subject has a long history, as long as mathematics itself, and is as actively researched today as at any time in the past. Number theory is not an organised theory in the usual sense, but a vast collection of individual topics and results, with some coherent subtheories and a long list of unresolved problems. The unit syllabus varies from year to year but it always includes some basic material on divisibility and congruences, quadratic residues and reciprocity law, multiplicative functions and sums of squares. Option parts include quadratic forms, prime number theory and arithmetic of elliptic curves.

Aims

At the end of the unit you will acquire a command of the basic tools of number theory as applicable to the investigation of congruences, arithmetic functions, Diophantine equations and beyond. In addition, you will become familiar with the underlying themes and current state of knowledge of several branches of Number Theory and its interaction with partner disciplines.

Syllabus

Topics covered will include:

1. Revision of the basic properties of the integers including the Euclidean algorithm.
2. Number-theoretic functions, especially the Möbius and Euler functions. Averages and maximum values.
3. Congruences, including the theorems of Fermat, Euler, and Lagrange, and computational applications. The RSA cryptosystem.
4. Primitive roots and the structure of the residues modulo m.
5. Polynomial congruences to prime powers. Hensel’s lemma and the p-adic numbers.
6. The quadratic residue symbols of Lagrange and Jacobi. Quadratic reciprocity.
7. The solution of quadratic equations in integers.
8. Introduction to one or more of the following topics, depending on time available: Diophantine approximation and transcendence, Dirichlet’s theorem on primes in arithmetic progressions, Diophantine equations and elliptic curves.

Relation to Other Units

This unit develops the number theory component of Level 1 Number Theory and Group Theory. The algebraic aspects of number theory are explored further in the partner Level 3 course Algebraic Number Theory.

Intended Learning Outcomes

After completing this unit successfully, students should be able to:

Understand and apply the basic properties of modular arithmetic so as to analyse the solubility of polynomial congruences and equations.

Estimate average and maximal values of basic arithmetic functions.

Exhibit some familiarity with the underlying themes and current state of knowledge of several branches of Number Theory and its interaction with partner disciplines.

Transferable Skills:

Ability to write coherent and logically sound arguments.

Assimilation and use of novel and abstract ideas.

Essay-writing.

Teaching Information

Lectures, homework exercises to be done by students.

Assessment Information

The final assessment mark for Number Theory is calculated from a 2 ½ -hour written examination in May/June consisting of FIVE questions. A candidate's best FOUR answers will be used for assessment. Calculators are NOT permitted to be used in this examination.

Reading and References

There is no recommended text for this course. Four useful references are:

Alan Baker, A concise introduction to the theory of numbers. Cambridge University Press, 1984. xiii+95 pp. ISBN: 0-521-24383-1; 0-521-28654-9

Ivan Niven, Herbert S. Zuckerman and Hugh L. Montgomery, An introduction to the theory of numbers. Fifth edition. John Wiley & Sons, Inc., 1991. xiv+529 pp. ISBN: 0-471-62546-9

H. E. Rose, A course in number theory. Second edition. Oxford Science Publications. The Clarendon Press, Oxford University Press, 1994. xvi+398 pp. ISBN: 0-19-853479-5; 0-19-852376-9

J. H. Silverman, A friendly introduction to number theory. Third edition. Prentice Hall, 2005, vii+434 pp. ISBN: 0-13-186137-9.