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Study information

# Algebraic Number Theory - 2023 entry

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 20 80 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 80 2 hours - Summer Exam Period All Exam mark, results released online
Coursework - problem sheet 1 7 1-3 pages of problems All Written comments on script and model solutions available
Coursework - problem sheet 2 7 1-3 pages of problems All Written comments on script and model solutions available
Coursework - problem sheet 3 6 1-3 pages of problems All Written comments on script and model solutions available

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
Written exam * Written Exam (2 hours) All August Ref/Def Period
Coursework 1 * Coursework 1  All August Ref/Def Period
Coursework 2 * Coursework 2

All

August Ref/Def Period
Coursework 3 * Coursework 3 All August Ref/Def Period

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

RE-ASSESSMENT NOTES

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%.

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

Reading list for this module:

Type Author Title Edition Publisher Year ISBN
Set Alaca, S. and Williams, K.S. Introductory Algebraic Number Theory Cambridge University Press 2004 978-0521540117
Set Frohlich, A. and Taylor, M.J. Algebraic Number Theory Cambridge University Press 1991 978-0521366649
Set Stewart, I. and Tall, D. Algebraic Number Theory and Fermat's Last Theorem 3rd Taylor and Francis 2002 978-1568811192
CREDIT VALUE ECTS VALUE 15 7.5
PRE-REQUISITE MODULES MTH3038
NQF LEVEL (FHEQ) AVAILABLE AS DISTANCE LEARNING 7 No Tuesday 10th July 2018 Thursday 26th January 2023
KEY WORDS SEARCH Algebraic Number Theory; Fermat's Last Theorem; Rings of Algebraic Integers

Please note that all modules are subject to change, please get in touch if you have any questions about this module.