Breadcrumb

Algebraic Number Theory 3 (MATH36205)

Academic Year:

Contents of this document:

Administrative Information

  1. Unit number and title: MATH36205 Algebraic Number Theory 3
  2. Level: H/6
  3. Credit point value: 20 credit points
  4. Year: 12/13
  5. First Given in this form: 2012 (Previously gives as a 10cp unit)
  6. Unit Organiser: Lynne Walling
  7. Lecturer: Dr Abhishek Saha
  8. Teaching block: 2
  9. Prerequisites: MATH 11511 (Number Theory & Group Theory), MATH 21800 (Algebra 2). MATH 30200 (Number Theory) is recommended but not necessary. Students may not take this unit with the corresponding Level 7 unit Algebraic Number Theory 34, or if they have already taken Algebraic Number Theory (MATH 31110).

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.

Relation to Other Units

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

Teaching Methods

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

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.

Assessment Methods

The final assessment mark for Algebraic Number Theory 3 is calculated from a 2½-hour written examination in May-June consisting of FIVE questions. A candidate's best FOUR answers will be used for assessment. Calculators are NOT permitted in this examination.

Award of Credit Points

Credit points for the unit are gained by passing the unit (i.e. getting a final assessment mark of 40 or over).

Transferable Skills

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

Texts

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

Syllabus

Number fields and their rings of integers.

Factoring in rings of integers.

Unique factorisation of ideals.

Ideal class groups.

Unit groups.

Norms, traces, and discriminants.

Applications to Diophantine equations.

Additonal topics may include:

Local fields; p-adic numbers; algorithmic aspects and applications; an introduction to adeles and proof of finiteness of class number in this language.