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.
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.
3 hr written examination or alternative assessment
Educational aims & objectives
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
2 hours of lectures per week, 1 hour of tutorials
Suggested reading list
Suggested reading/resources (link to My Reading Lists)