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Algebraic Number Theory 34 (MATHM6205)

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Administrative Information

  1. Unit number and title: MATHM6205 Algebraic Number Theory 34
  2. Level: M/7
  3. Credit point value: 20 credit points
  4. Year: 12/13
  5. First Given in this form: 2012
  6. Unit Organiser: Abhishek Saha
  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 6 unit MATH36205 (Algebraic Number Theory 3), or if they have already taken MATH 31110 (Algebraic Number Theory).

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).

Teaching Methods

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.

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.

Assessment Methods

Formative assessment will be provided by problem sheets with questions that will be set and marked through the course.


The final assessment mark will be based on:

  • 80% of the mark will come 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.
  • 20% of the mark will come from a written project and short presentation, involving the independent study, an awareness of a broader literature and a critical evaluation of how the basic ideas may be further developed. Successful completion of the project component will require evidence of substantially greater depth of understanding of the area as a whole than that required for the corresponding Level 6 unit.

Award of Credit Points

Credit points for the unit are gained by passing the unit (i.e. getting a final assessment mark of 50 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.