## Algebraic Number Theory

 Unit code: MATH62132 Credit Rating: 15 Unit level: Level 6 Teaching period(s): Semester 2 Offered by School of Mathematics Available as a free choice unit?: N

None

#### Aims

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

#### 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.

#### Learning outcomes

Learning outcomes.  On successful completion of this course unit students will

• Be able to calculate the ring of integers of simple number fields.
• Understand how the class group controls the failure of unique factoristion.
• Be able to calculate the class group in simple examples.
• Be able to factorise ideals into products of prime ideals.
• Solve some non-linear diophantine equations.

#### Assessment methods

• Other - 10%
• Written exam - 90%

#### 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

• Stewart and Tall, Algebraic Number Theory and Fer- mat’s Last Theorem.
• Jarvis, Algebraic Number Theory.
• Neukirch, Algebraic Number Theory.

#### 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.

#### Study hours

• Lectures - 22 hours
• Tutorials - 11 hours
• Independent study hours - 117 hours

#### Teaching staff

Daniel Loughran - Unit coordinator