Course Catalogue

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This course introduces basic ideas of modern number theory, using earlier knowledge including abstract algebra.

17 x 1 hr lectures and 6 x 1 hr tutorials in a semester

Mandatory Entry Requirements

3H Algebra (MATHS4072)

3H Methods in Complex Analysis (MATHS4076)

Recommended Entry Requirements

Assessment

90% Examination, 10% Coursework.

Reassessment

In accordance with the University's Code of Assessment reassessments are normally set for all courses which do not contribute to the honours classifications. For non honours courses, students are offered reassessment in all or any of the components of assessment if the satisfactory (threshold) grade for the overall course is not achieved at the first attempt. This is normally grade D3 for undergraduate students, and grade C3 for postgraduate students. Exceptionally it may not be possible to offer reassessment of some coursework items, in which case the mark achieved at the first attempt will be counted towards the final course grade. Any such exceptions are listed below in this box.

Reassessments are normally available for all courses, except those which contribute to the Honours classification. For non Honours courses, students are offered reassessment in all or any of the components of assessment if the satisfactory (threshold) grade for the overall course is not achieved at the first attempt. This is normally grade D3 for undergraduate students and grade C3 for postgraduate students. Exceptionally it may not be possible to offer reassessment of some coursework items, in which case the mark achieved at the first attempt will be counted towards the final course grade. Any such exceptions for this course are described below. 

The main topics will include review of basic material on congruences, Fermat's Little Theorem and Euler's Theorem, quadratic reciprocity, continued fractions and Pell's equation, a basic introduction to algebraic number theory including quadratic and cyclotomic number fields, rings of integers, factorisation theory and diophantine problems, arithmetic functions, introduction to analytic number theory including asymptotic estimates for the distribution of primes and the relationship of these topics with the Riemann Zeta function and the Riemann Hypothesis.

By the end of this course students will be able to:

Demonstrate knowledge of the central definitions and facts of number theory and use these to solve problems of a numerical or logical nature. Topics covered will include a subset of:

1. Quadratic reciprocity, continued fractions and Pell's equation.

2. Algebraic number theory including quadratic and cyclotomic number fields.

3. Dirichlet series and the distribution of prime numbers.

4. Riemann Zeta function and the Riemann Hypothesis.

5. The Prime Number Theorem.

Students must submit at least 75% by weight of the components (including examinations) of the course's summative assessment.