# Mathematics / Applied Mathematics MSc

# 5E: Number Theory MATHS5074

**Academic Session:**2019-20**School:**School of Mathematics and Statistics**Credits:**10**Level:**Level 5 (SCQF level 11)**Typically Offered:**Semester 1**Available to Visiting Students:**No**Available to Erasmus Students:**No

#### Short Description

This course introduces basic ideas of modern number theory, using earlier knowledge including abstract algebra. Students taking this version will cover some material by directed reading and additional examples.

#### Timetable

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

#### Requirements of Entry

Mandatory Entry Requirements

3H Algebra (MATHS4072)

Recommended Entry Requirements

4H Galois Theory(MATHS4105)

#### Excluded Courses

4H: Number Theory (MATHS4108)

#### Assessment

Assessment

90% Examination, 10% Coursework.

Reassessment

Resit opportunities for MSc students.

**Main Assessment In:** April/May

#### Course Aims

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.

#### Intended Learning Outcomes of Course

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

a) solve congruence problems using results from the course;

b) use continued fractions to find solutions to Pell's equation;

c) exploit the algebraic structure of quadratic number fields and cyclotomic fields to solve Diophantine problems;

d) work with arithmetic functions using convolution and MÃ¶bius inversion;

e) state the Prime Number Theorem and deduce basic results from it.;

f) recall results on general number fields and their rings of integers, including cyclotomic fields.

#### Minimum Requirement for Award of Credits

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