| Unit name | Algebraic Number Theory 4 |
|---|---|
| Unit code | MATHM6205 |
| Credit points | 20 |
| Level of study | M/7 |
| Teaching block(s) |
Teaching Block 2 (weeks 13 - 24) |
| Unit director | Dr. Bouyer |
| Open unit status | Not open |
| Units you must take before you take this one (pre-requisite units) |
MATH21800 Algebra 2 MATH30200 Number Theory, and MATH33300 Group Theory are recommended but not necessary. |
| Units you must take alongside this one (co-requisite units) |
MATHM2700 Galois Theory is recommended but not necessary. |
| Units you may not take alongside this one | |
| School/department | School of Mathematics |
| Faculty | Faculty of Science |
Unit Aims
Unit Description
Integers and rational numbers are the first numbers we encounter, and as such they are, in a way the easiest numbers to think with. So when we come across an equation, say for example the one that arises from Pythagoras Theorem, we can be tempted to ask: which integers solves these equations, and can we find all of them? Trying to find all integer solutions to a given equation is called solving Diophantine equations, and Number Theory is the study of Diophantine equations.
Broadly speaking Algebraic Number Theory tries to solve number theory questions by using tools and techniques from abstract algebra. In this course we will focus on number fields (extensions of the rationals), their ring of integers (the analogue of the integers) and their various properties. We will see that unique factorisation doesn't work in number fields and therefore we will introduce ideals (an analogue of numbers) to go around that problem. By the end of the units, all these tools will be used to solve various Diophantine equations.
Relation to Other Units
The course build on the material of MATH21800 Algebra 2 and has relations to MATHM2700 Galois Theory. The material is complementary to that of MATHM0007 Analytic Number Theory.
Students may not take this unit with the corresponding Level 6 unit MATH36205 Algebraic Number Theory 3.
Learning Objectives
Students who successfully complete the unit should be able to:
By pursuing an individual project on a more advanced topic students should have:
Transferable Skills
Using an abstract framework to better understand how to attack a concrete problem.
The unit will be taught through a combination of
80% Timed, open-book examination 20% Coursework.
Raw scores on the examinations will be determined according to the marking scheme written on the examination paper. The marking scheme, indicating the maximum score per question, is a guide to the relative weighting of the questions. Raw scores are moderated as described in the Undergraduate Handbook.
If this unit has a Resource List, you will normally find a link to it in the Blackboard area for the unit. Sometimes there will be a separate link for each weekly topic.
If you are unable to access a list through Blackboard, you can also find it via the Resource Lists homepage. Search for the list by the unit name or code (e.g. MATHM6205).
How much time the unit requires
Each credit equates to 10 hours of total student input. For example a 20 credit unit will take you 200 hours
of study to complete. Your total learning time is made up of contact time, directed learning tasks,
independent learning and assessment activity.
See the Faculty workload statement relating to this unit for more information.
Assessment
The Board of Examiners will consider all cases where students have failed or not completed the assessments required for credit.
The Board considers each student's outcomes across all the units which contribute to each year's programme of study. If you have self-certificated your absence from an
assessment, you will normally be required to complete it the next time it runs (this is usually in the next assessment period).
The Board of Examiners will take into account any extenuating circumstances and operates
within the Regulations and Code of Practice for Taught Programmes.
