Algebraic Number Theory - 2023 entry

Algebraic Number Theory is the study of algebraic numbers, and is a topic at the forefront of research in modern pure mathematics. The topic grew from a desire to prove Fermat’s Last Theorem, conjectured by Pierre de Fermat, in 1637, and proved by Andrew Wiles, in 1995. This module introduces and examines classes of algebraic objects, including algebraic number fields, rings of algebraic integers, and the set ideals in a ring of algebraic integers.

Towards the end of the module, you will learn about the factorisation of ideals in rings of algebraic integers. The crowning glory of this module is the examination of the ideal class group of an algebraic number field. This object measures the extent to which a ring of algebraic integers fails to be a principal ideal domain.

Pre-requisite Module: MTH3038 Galois Theory, or equivalent

The aim of this module is to expose you to an important area of modern pure mathematics, namely the theory of algebraic number fields and their rings of integers. This underlies much contemporary research in number theory and arithmetic geometry, as well as finding applications in areas such as cryptography.

On successful completion of this module, you should be able to:

Module Specific Skills and Knowledge:

1 Give definitions of the mathematical objects introduced in this module, such as algebraic numbers, algebraic number fields, algebraic integers, ideals in a ring of integers, the ideal class group;

2 Give definitions of functions, invariants, and other quantities associated with the objects mentioned in 1, such as norms and traces of algebraic numbers, rational and integral bases of algebraic number fields, discriminants of bases of number fields, norms of ideals, and the class number;

3 Perform computations related to the mathematical objects introduced in this module;

4 State and prove theorems concerning the mathematical objects introduced in this module;

Discipline Specific Skills and Knowledge:

5 Comprehend the role of algebraic methods in the systematic study of arithmetic problems;

6 Be proficient at learning an advanced pure mathematical topic via the traditional definition-theorem-proof style of exposition;

Personal and Key Transferable/ Employment Skills and Knowledge:

7 Display enhanced problem-solving skills and ability to construct and comprehend rigorous mathematical arguments.

- Algebraic Number Fields: degree, embeddings into the complex numbers; rings of integers; integral bases; discriminants;

- Ideals;

- Failure of unique factorisation into irreducibles;

- Uniqueness of factorisation into prime ideals;

- Equivalence of ideals;

- The Ideal Class Group;

- Minkowski's Theorem;

- Proof of finiteness of the class group;

- Calculation of the class group in particular cases;

- Application to Diophantine Equations.

*Please refer to reassessment notes for details on deferral vs. Referral reassessment

Deferrals: Reassessment will be by coursework and/or written exam in the deferred element only. For deferred candidates, the module mark will be uncapped. 

Referrals: Reassessment will be by a single written exam worth 100% of the module only. As it is a referral, the mark will be capped at 50%. 

ELE – http://vle.exeter.ac.uk

Reading list for this module: