Algebraic Number Theory

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 Algebra 2 (MATH21800) and has relations to Galois Theory (MATHM2700).  The material is complementary to that of Analytic Number Theory (MATHM0007).

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.

Transferable skills

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

Syllabus

Number fields and their rings of integers.

Norms, traces, and discriminants.

Unique factorisation of ideals.

Factoring ideals in rings of integers.

Ideal class groups.

Unit groups.

Applications to Diophantine equations.

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

Unit code: MATHM6205
Level of study: M/7
Credit points: 20
Teaching block (weeks): 2 (13-24)
Lecturer: Dr Florian Bouyer

Pre-requisites

MATH21800 Algebra 2.

MATH30200 Number Theory, MATH33300 Group Theory and MATHM2700 Galois Theory are strongly recommended but not necessary. Students may not take this unit with the corresponding Level 6 unit MATH36205 Algebraic Number Theory 3.

Co-requisites

None

Methods of teaching

Lectures, including examples and revision classes, supported by lecture notes with problem sets and model solutions.

Methods of Assessment

The pass mark for this unit is 50.

The final mark is calculated as follows:

  • 80% Exam
  • 20% Coursework

NOTE: Calculators are NOT allowed in the examination.

For information resit arrangements, please see the re-sit page on the intranet.

Please use these links for further information on relative weighting and marking criteria.

Further exam information can be found on the Maths Intranet.

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