Unit information: Algebraic Number Theory 4 in 2019/20

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
Pre-requisites	MATH21800 Algebra 2 MATH30200 Number Theory, and MATH33300 Group Theory are recommended but not necessary.
Co-requisites	MATHM2700 Galois Theory is recommended but not necessary.
School/department	School of Mathematics
Faculty	Faculty of Science

Description including Unit Aims

Unit Aims

• To gain an understanding and appreciation of algebraic number theory.
• To become familiar with concepts such as number fields, rings of integers and ideals.
• To become comfortable in using tools and techniques from algebraic number theory to solve Diophantine equations.

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.

Intended Learning Outcomes

Learning Objectives

Students who successfully complete the unit should be able to:

• Understand and clearly define number fields and their ring of integers, in particular quadratic number fields and cyclotomic number fields;
• Define, describe and analyse more advanced concepts such as ideals, ideal classes, unit groups, norms, traces and discriminant;
• Find the factorisation of ideals, the ring of integers, the class number and ideal class group of a number field;
• Solve certain Diophantine equations by applying tools from the course.

By pursuing an individual project on a more advanced topic students should have:

• developed an awareness of a broader literature;
• gained an appreciation of how the basic ideas may be further developed;
• learned how to assimilate material from several sources into a coherent document.

Transferable Skills

Using an abstract framework to better understand how to attack a concrete problem.

Teaching Information

Lectures, including examples and revision classes, supported by lecture notes with problem sets and model solutions. Self-study with directed reading based on recommended material.

Assessment Information

80% Examination and 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.

Reading and References

Further

• Saban Alaca and Kenneth S. Williams, Introductory Algebraic Number Theory, Cambridge University Press, 2003
• Daniel A. Marcus, Number Fields, Springer, 1977
• Ian Stewart and David Tall, Algebraic Number Theory and Fermat’s Last Theorem, AK Peters, 2002