Unit information: Algebraic Number Theory 4 in 2018/19

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 Group Theory (MATH 33300) are recommended but not necessary. Students may not take this unit with the corresponding Level 6 unit MATH36205 (Algebraic Number Theory 3).
Co-requisites	None
School/department	School of Mathematics
Faculty	Faculty of Science

Description including Unit Aims

Unit aims

The aims of this unit are to enable students to gain an understanding and appreciation of algebraic number theory and familiarity with the basic objects of study, namely number fields and their rings of integers. In particular, it should enable them to become comfortable working with the basic algebraic concepts involved, and to see applications of the theory to Diophantine equations.

General Description of the Unit

Algebraic Number Theory is a major branch of Number Theory (alongside Analytic Number Theory) which studies the algebraic properties of algebraic numbers, and number fields. The unit will provide an introduction to algebraic number theory. The unit will focus on algebraic number fields and their rings of integers (how to generalise the usual integers), ideals (how to factorise and work with them), units and the ideal class group. With all these tools, the unit will explore applications to solving certain Diophantine equations.

Relation to Other Units

The course build on the material of Algebra 2 (Math 21800) and has relations to Galois Theory (Math M2700). The material is complementary to that of Analytic Number Theory (Math M0007).

Additional unit information can be found at http://www.maths.bristol.ac.uk/study/undergrad/current_units/index.html

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

Reading and references are available at http://www.maths.bristol.ac.uk/study/undergrad/current_units/index.html