Module description
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.
Prerequisites
Before taking this module you are advised to have knowledge and experience of the topics covered in the syllabus for 6CCM326A/7CCM326B Galois Theory, 6CCM350A Rings & Modules and 7CCMMS15 Advanced Algebra.
Assessment details
Assessment
2 hour written examination.
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
Teaching pattern
Two hours of lectures and one hour of tutorial per week throughout the term
Suggested reading list
Indicative reading list - link to Leganto system where you can search with module code for lists