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
Credit level: 7 Credit Value: 15
Mathematics BSc/MSci (or with Year Abroad)
Mathematics and Physics BSc/MSci (or with Year Abroad)
MSc Theoretical Physics
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
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.
2 hours of lectures per week. 1 further hour will be used for lecture or tutorial as required
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.
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|3 hr written examination
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.
Suggested reading/resources (link to My Reading Lists)
15 August 2018