The Classification and Structure of Locally-compact Topological Fields

Tuesday, May 3, 2016 - 12:30pm
312 MSB
Robert Biggs (Advisor: Prof. William Banks)

This presentation covers the classification of local fields, which are topological fields whose topology is both Hausdorff and locally-compact. These spaces are special in that they carry enough structure to investigate analogs of analysis over fields other than the real number field or the complex numbers. We begin by studying the module of a local field and investigate the consequences of its properties. We then investigate topological vector spaces over local fields. Then, using the characterization of absolute values, on the rational numbers, we classify all archimedean local fields. From there, we move on to investigate non-archimedean local fields and the interesting algebraic properties they possess which is absent in real or complex analysis. We finish by classifying local fields of positive characteristic and explore extensions of the p-adic rationals