MSc Pure Mathematics and Mathematical Logic

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.
 

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

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.

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

Single piece of take-home coursework; weighting 20%

End of semester examination: weighting 80%

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)

This course unit detail provides the framework for delivery in 20/21 and may be subject to change due to any additional Covid-19 impact.  

Please see Blackboard / course unit related emails for any further updates.