# Mathematics

## MTHM028 - Algebraic Number Theory (2019)

MODULE TITLE CREDIT VALUE Algebraic Number Theory 15 MTHM028 Prof Andreas Langer (Coordinator)
DURATION: TERM 1 2 3
DURATION: WEEKS 0 11 0
 Number of Students Taking Module (anticipated) 15
DESCRIPTION - summary of the module content

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

AIMS - intentions of the module

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.

INTENDED LEARNING OUTCOMES (ILOs) (see assessment section below for how ILOs will be assessed)

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.

SYLLABUS PLAN - summary of the structure and academic content of the module

- 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.

LEARNING AND TEACHING
LEARNING ACTIVITIES AND TEACHING METHODS (given in hours of study time)
 Scheduled Learning & Teaching Activities Guided Independent Study Placement / Study Abroad 33 117 0
DETAILS OF LEARNING ACTIVITIES AND TEACHING METHODS
 Category Hours of study time Description Scheduled Learning and Teaching Activities 33 Lectures Guided Independent Study 117 Assessment preparation; private study

ASSESSMENT
FORMATIVE ASSESSMENT - for feedback and development purposes; does not count towards module grade
Form of Assessment Size of Assessment (e.g. duration/length) ILOs Assessed Feedback Method
Coursework – Problem Sheets 1, 2, 3 1-3 pages of problems All Written comments on script and model solutions available

SUMMATIVE ASSESSMENT (% of credit)
 Coursework Written Exams Practical Exams 0 100 0
DETAILS OF SUMMATIVE ASSESSMENT
Form of Assessment % of Credit Size of Assessment (e.g. duration/length) ILOs Assessed Feedback Method
Written Exam – Closed Book 100 2 hours All Written/verbal on request

DETAILS OF RE-ASSESSMENT (where required by referral or deferral)
Original Form of Assessment Form of Re-assessment ILOs Re-assessed Time Scale for Re-reassessment
All Above Written Exam (100%) All August Ref/Def Period

RE-ASSESSMENT NOTES

If a module is normally assessed entirely by coursework, all referred/deferred assessments will normally be by assignment.

If a module is normally assessed by examination or examination plus coursework, referred and deferred assessment will normally be by examination. For referrals, only the examination will count, a mark of 50% being awarded if the examination is passed. For deferrals, candidates will be awarded the higher of the deferred examination mark or the deferred examination mark combined with the original coursework mark.

RESOURCES
INDICATIVE LEARNING RESOURCES - The following list is offered as an indication of the type & level of
information that you are expected to consult. Further guidance will be provided by the Module Convener