Unit information: Algebraic Number Theory 4 in 2014/15

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. Saha
Open unit status	Not open
Pre-requisites	MATH 11511 (Number Theory & Group Theory), MATH 21800 (Algebra 2). MATH 30200 (Number Theory), Group Theory (MATH 33300) and Galois Theory (MATH M2700) are recommended but not necessary. Students may not take this unit with the corresponding Level 6 unit MATH36205 (Algebraic Number Theory 3), or if they have already taken MATH 31110 (Algebraic Number Theory).
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, to appreciate the failure of unique factorisation in general, 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 – in particular the factorization of algebraic integers and ideals – in a setting in which familiar features of the (usual) integers, such as unique factorization, need not hold. The unit will provide an introduction to algebraic number theory, focussing on algebraic number fields and their rings of integers, ideals and factorization, units and the ideal class group, and will explore some applications to Diophantine equations.

In addition, students will have the opportunity to develop an awareness of a broader literature and gain an appreciation of how the basic ideas may be further developed through an individual project.

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).

Further information is available on the School of Mathematics website: http://www.maths.bris.ac.uk/study/undergrad/

Intended Learning Outcomes

Learning Objectives

Students who successfully complete the unit should be able to:

• clearly define, describe and analyse standard examples of algebraic number fields and their rings of integers;
• appreciate and comment critically on the variety of these examples, and especially the failure of unique factorisation in general;
• clearly define, describe and analyse more advanced concepts such as ideals, ideal classes, unit groups, norms, traces and discriminants;
• perform algebraic manipulations with these, especially as required for applications to Diophantine equations.

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

Lecture notes and handouts will be provided covering all the main material.

The following supplementary texts provide additional background reading:

• Algebraic Number Theory and Fermat’s Last Theorem, I. Stewart and D. Tall, AK Peters, 2002
• Introductory Algebraic Number Theory, S. Alaca and K.S. Williams, CUP, 2003
• Number Fields, D. Marcus, Springer, 1977