MA220-7-AU-CO:
Number Theory
2022/23
Mathematics, Statistics and Actuarial Science (School of)
Colchester Campus
Autumn
Postgraduate: Level 7
Current
Thursday 06 October 2022
Friday 16 December 2022
15
30 April 2022
Requisites for this module
(none)
(none)
(none)
(none)
(none)
This module gives a broad introduction to the mathematics of the integers, primes and modular arithmetic, including both classical and modern viewpoints.
Number theory encompasses some of the most classical and important topics in mathematics, stemming from the study of integers, Diophantine equations, prime numbers and modular arithmetic. As well as introducing each of these, in this module it will be demonstrated how techniques from a range of mathematical disciplines such as algebra and geometry can be applied to these topics.
On successful completion of the module, students will:
1. Have a comprehensive understanding of the theory of low-degree Diophantine equations and congruence equations, and the tools used in their study, including quadratic reciprocity, Fermat’s Little Theorem and the failure of its converse (Carmichael Numbers), and the relationship with research-level concepts such as the p-adic numbers.
2. Be able to creatively develop and critically evaluate number-theoretic statements, including both practical calculations and broad general statements.
3. Have a comprehensive understanding of number systems such as Dedekind domains, rings of integers such as the Gaussian integers, and related number rings, such as algebraic numbers, algebraic integers and transcendental numbers.
4. Develop practical skill with tools relating to number systems, both as objects of study in their own right, and as tools in areas such as Diophantine approximation
5. Have a critical awareness and ability to evaluate open problems in number theory and the modern theoretical tools being used in their research.
Syllabus:
Introduction to Diophantine equations – Pythagorean triples, solutions via points on rational curves.
Integer and modular arithmetic – Fundamental Theorem of Arithmetic; applications of the Euclidean algorithm; modular arithmetic; Chinese Remainder Theorem; the Euler phi function and its properties.
Quadratic reciprocity – Quadratic residues; Euler's criterion; relation to sums of squares.
Diophantine approximation and continued fractions – algebraic and transcendental numbers, Diophantine approximation, solution to Pell's equation.
Gaussian integers – The norm and its properties; Gaussian primes; sums of squares.
Elements of ring theory: Ring-theoretic properties of algebraic numbers, algebraic integers, Gaussian integers and related number rings.
Asymptotics and distribution of primes – asymptotic notation, statement of the Prime Number Theorem.
(Level 7) Algebraic number theory: Unique factorisation of ideals, Dedekind domains, fractional ideals, class numbers.
(Level 7) p-adic numbers: p-adic norm, analytical and topological properties, Henselian lifting.
3 contact hours per week. This will consist mainly of lectures, with 1 class every 1 or 2 weeks. 3 revision lectures to be given in the summer term.
Level 7 – 5 additional lectures, spaced across the term (content assessed in Summer exam).
This module does not appear to have a published bibliography for this year.
Assessment items, weightings and deadlines
| Coursework / exam |
Description |
Deadline |
Coursework weighting |
| Coursework |
Assignment |
|
|
| Exam |
Main exam: In-Person, Open Book (Restricted), 120 minutes during Summer (Main Period)
|
| Exam |
Reassessment Main exam: In-Person, Open Book (Restricted), 120 minutes during September (Reassessment Period)
|
Exam format definitions
- Remote, open book: Your exam will take place remotely via an online learning platform. You may refer to any physical or electronic materials during the exam.
- In-person, open book: Your exam will take place on campus under invigilation. You may refer to any physical materials such as paper study notes or a textbook during the exam. Electronic devices may not be used in the exam.
- In-person, open book (restricted): The exam will take place on campus under invigilation. You may refer only to specific physical materials such as a named textbook during the exam. Permitted materials will be specified by your department. Electronic devices may not be used in the exam.
- In-person, closed book: The exam will take place on campus under invigilation. You may not refer to any physical materials or electronic devices during the exam. There may be times when a paper dictionary,
for example, may be permitted in an otherwise closed book exam. Any exceptions will be specified by your department.
Your department will provide further guidance before your exams.
Overall assessment
Reassessment
Module supervisor and teaching staff
Dr Alastair Litterick, email: a.litterick@essex.ac.uk.
Dr Alastair Litterick
a.litterick@essex.ac.uk
No
No
Yes
Dr Rachel Quinlan
National University of Ireland, Galway
Senior Lecturer in Mathematics
Available via Moodle
Of 38 hours, 38 (100%) hours available to students:
0 hours not recorded due to service coverage or fault;
0 hours not recorded due to opt-out by lecturer(s), module, or event type.
Disclaimer: The University makes every effort to ensure that this information on its Module Directory is accurate and up-to-date. Exceptionally it can
be necessary to make changes, for example to programmes, modules, facilities or fees. Examples of such reasons might include a change of law or regulatory requirements,
industrial action, lack of demand, departure of key personnel, change in government policy, or withdrawal/reduction of funding. Changes to modules may for example consist
of variations to the content and method of delivery or assessment of modules and other services, to discontinue modules and other services and to merge or combine modules.
The University will endeavour to keep such changes to a minimum, and will also keep students informed appropriately by updating our programme specifications and module directory.
The full Procedures, Rules and Regulations of the University governing how it operates are set out in the Charter, Statutes and Ordinances and in the University Regulations, Policy and Procedures.