A typical example of a local field is constructed as the completion of the field of rationals Q with respect to a fixed valuation on Q. I will talk about a classification theorem of local fields due to Ostrowski and see that the metric defined on a non-archimedean local field looks different from the Euclidean metric. As an application, we will discuss when a given quadratic form has a rational solution. If time permits, we would like to mention Hilbert symbols and quaternion algebras.
Reference: Serre, Local fields, Springer
In this talk, I discuss an application of category theory to rings and singularities. This will provide subtle information of commutative rings of positive characteristic. Our target is a singularity called F-pure. I also introduce new definition of F-purity.
In this talk, I will show how to compute lc-thresholds of a singularity or how to prove that certain singularity is a rational singularity or not by characteristic p method by the examples of rational double points and jet schemes.