Project research area: Pure mathematics (Number Theory, Algebra)DescriptionThis project will explore results in algebraic number theory that build on but lie beyond what is covered in Number Theory III.We start with Minkowski's Geometry of numbers (see Chapter II in [3] or Chapter 7 in [1]), which is used 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 existence of fundamental units in real quadratic number fields (and the corresponding fact that any unit is plus or minus a power of the fundamental unit). There is a story (and some historical records to back it up; see [5]) that Dirichlet found some important ingredient of his proof during an Easter Mass in the Sistine Chapel, although he did not use Minkowski's theory, as that was developed later. After this, the project can be taken in several different individual directions, depending on interest. One option is to study Dedekind zeta functions (parts of Chapter 10 in [1], Chapter 7 in [2]), which are generalisations of the Riemann zeta function from \(\mathbb{Q}\) to arbitrary number fields. This leads to the Analytic class number formula, which relates the residue of the Dedekind zeta function at its pole of a number field to several invariants such as the class number, the discriminant and an invariant called the regulator. Other possible directions for further study include:
Mode of Operation and Evidence of LearningThe project will revolve around learning through reading with focus on the underlying theory, mathematical rigour, and the development of deep conceptual understanding. Students will demonstrate their understanding presenting the material in their own way, exploring examples (existing or new) and theoretical applications of the material, and clearly communicating it in both written and oral formats.PrerequisitesNumber Theory III. In addition, Galois Theory will be helpful for certain optional directions of individual study, but is not essential for the project.Resources
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email: Alexander Stasinski