# MMath Mathematics with Financial Mathematics

Year of entry: 2023

## Course unit details:Algebraic Number Theory

Unit code MATH42132 15 Level 4 Semester 2 Department of Mathematics No

### Overview

Number theory is arguably one of the oldest and most fascinating bran- ches of mathematics. This fascination stems from the fact that there are a great many theorems concerning the integers, which are extremely simple to state, but turn out to be rather hard to prove.

The fundamental objects in algebraic number theory are finite field extensions of Q; so-called number fields. To a number field k one associates a ring O_k  called its ring of integers. This ring behaves in some respects like the usual ring of integers Z, however many well know properties of Z do not pass over; the most important being that the fundamental theorem of arithmetic can fail in O_k .

The main focus of this course is on the failure of the unique factorisation. We also give a number applications to the study of certain diophantine equations.

### Pre/co-requisites

Unit title Unit code Requirement type Description
MATH32011 Pre-Requisite Compulsory

Students are not permitted to take, for credit, MATH42132 in an undergraduate programme and then MATH62132 in a postgraduate programme at the University of Manchester, as the courses are identical

### Aims

To show how tools from algebra can be used to solve problems in number theory.

### Learning outcomes

On successful completion of this course unit students will be able to:

• define the basic notions of algebraic number theory, such as algebraic numbers and integers, conjugates, number fields, rings of integers, norm, trace and discriminant, fractional ideals, class groups and lattices,
• describe the additive and multiplicative structure of a number field and it’s the ring of integers using the proper algebraic terminology,
• perform basic computations with algebraic integers in a simple number field, such as addition and multiplication, finding inverses and computing the minimal polynomial,
• identify the ring of integers and the discriminant of simple examples, such as quadratic and cyclotomic fields, and justify the identification,
• summarise a procedure to factorise prime numbers into prime ideals of a ring of integers and apply it in the case of simple number fields, such as quadratic fields,
• re-formulate statements concerning the existence of certain algebraic integers in terms of lattice points and apply Minkowski’s first theorem to prove them,
• compute class numbers and class groups of simple number fields, such as quadratic fields,
• solve simple Diophantine equations using factorisations of algebraic integers and ideals.

### Syllabus

Fields and rings [2 lectures]
- Review of required tools from the theory of fields and rings
- Field extensions, ideals, maximal ideals, prime ideals
- Euclidean domain  => PID  => UFD  => integral domain

Number fields [2 lectures]
- Definitions and basic examples
- Embeddings into the real and complex numbers
- Field norms and trace

Rings of integers [4 lectures]
- Integral closures
- Definitions and basic properties
- Discriminants
- Calculation for quadratic field extensions and cyclotomic fields

Unique factorisation of ideals [4 lectures]
- Prime ideals in rings of integers of number fields
- Unique factorisation into prime ideals

Geometry of numbers [4 lectures]
- Lattices
- The Minkowski bound

Failure of unique factorisation [4 lectures]
- Examples
- Definition and finiteness of the class group

Applications [2 lectures]
- Applications to non-linear Diophantine equations
- Some cases of Fermat’s last theorem

### Assessment methods

Method Weight
Other 20%
Written exam 80%

Coursework: Single piece of take-home coursework, weighting 20%

End of semester examination: weighting  80%

### Feedback methods

Feedback tutorials will provide an opportunity for students’ work to be discussed and provide feedback on their understanding. Coursework or in-class tests (where applicable) also provide an opportunity for students to receive feedback. Students can also get feedback on their understanding directly from the lecturer, for example during the lecturer’s office hour.

The course notes are self-contained, no additional reading is required. The course is based on the following books:

Stewart and Tall, Algebraic Number Theory and Fermat's Last Theorem (recommended)
Jarvis, Algebraic Number Theory (recommended)
Marcus, Algebraic Number Fields (recommended)
Neukirch, Algebraic Number Theory (further)

### Study hours

Scheduled activity hours
Lectures 12
Tutorials 12
Independent study hours
Independent study 126

### Teaching staff

Staff member Role
Vandita Patel Unit coordinator

The independent study hours will normally comprise the following. During each week of the taught part of the semester:

·         You will normally have approximately 75-120 minutes of video content. Normally you would spend approximately 2.5-4 hrs per week studying this content independently

·         You will normally have exercise or problem sheets, on which you might spend approximately 2-2.5hrs per week

·         There may be other tasks assigned to you on Blackboard, for example short quizzes, short-answer formative exercises or directed reading

·         In some weeks you may be preparing coursework or revising for mid-semester tests

Together with the timetabled classes, you should be spending approximately 9 hours per week on this course unit.

The remaining independent study time comprises revision for and taking the end-of-semester assessment.

The above times are indicative only and may vary depending on the week and the course unit. More information can be found on the course unit’s Blackboard page.