Modules

7CCMMS03 Algebraic Number Theory

7CCMMS03T (MSc Programme)/ 7CCMMS03U (MSci Programme)

Lecturer:  Professor Nicholas Shepherd-Barron

Advice/help for this module is also available from: tbc

Semester:  2

Credit level:  7       Credit Value: 15

Programmes

Mathematics BSc/MSci (or with Year Abroad)
Mathematics and Physics BSc/MSci (or with Year Abroad)

MSc Mathematics

MSc Theoretical Physics

Aims

To give a thorough understanding of the `arithmetic' of number fields (finite extensions of Q) and their rings of integers, making use of abstract algebra. We shall note the analogies and differences between this arithmetic and that of Q and Z (e.g. unique factorisation may not hold). This motivates the study of ideals of the ring of integers, the class group and units. Concrete examples will illustrate the theory. This course provides a foundation for studies in modern (algebraic) number theory and is an essential ingredient of some other areas of algebra and arithmetic geometry

Syllabus

Polynomials and field extensions (brief reminders and terminology). Number fields. Norm, trace and characteristic polynomial. The ring of integers, integral bases, discriminant. Quadratic fields. Cyclotomic fields. Non-unique factorisation of elements, ideals, unique factorisation of ideals, norms of ideals, class group. Lattices, Minkowski's Theorem, computation of the class group. Extra topics (as time allows): Applications to Diophantine equations, Units, Dirichlet's Unit Theorem.

Teaching arrangements

2 hours of lectures per week. 1 further hour will be used for lecture or tutorial as required

Prerequisites

Normally you should have taken Rings and Modules (6CCM350a/7CCM350b) and be familiar with the elementary theory of field extensions (degree, minimal polynomials and algebraicity, embeddings eg as contained in the early part of the syllabus for Galois Theory (6CCM326a/7CCM326b)). If either condition is not met, the lecturer must be consulted before you register for the course.

Summative assessment
TypeWeightMarking Model
3 hr written examination 100% Model 2

Formative assessment

Exercise sheets will be distributed weekly in lectures. Full solutions will be provided. Doing the exercises and attending lectures and tutorials are essential to following the course. For this reason they are compulsory.