DescriptionThis project will explore results and methods of algebraic number theory that build on but lie beyond what is covered in Number Theory III.We start with Minkowski's Geometry of numbers (Chapter 7 in Jarvis's book), which is about lattices (for us, certain discrete subgroups of real vector spaces) and Minkowski's theorem on the existence of lattice points. We then use this to prove the Minkowski bound, as well as another major theorem in algebraic number theory: Dirichlet's Unit Theorem. The latter is a generalisation, to arbitrary number fields, of the main theorem on fundamental units in real quadratic number fields. After this, the project can be taken in several different individual directors, depending on interest. One possibility is to study Dedekind zeta functions (parts of Chapter 10 in Jarvis's book, Chapter 7 in Marcus's book), which are generalisations of the Riemann zeta functions from \(\mathbb{Q}\) to arbitrary number fields. This leads to the Analytic class number formula, which relates the residue of the Dedekind zeta function of a number field to several invariants such as the class number, the discriminant and an invariant called the regulator. PrerequisitesA solid knowledge of Number Theory III.Resources
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email: Alexander Stasinski