DescriptionThe aim of this project is to study number theory in so-called local fields. The study of local fields is a cornerstone of modern number theory. One of the first examples of local fields is the field of $p$-adic numbers $\mathbb{Q}_p$, which are easier to deal with than number fields, because in a certain sense $p$-adic fields only have one prime. After introducing $p$-adic numbers and general local fields, we look at their basic properties and examples. We then move on to extensions of local fields (e.g. finite Galois extensions), which is where much of the action takes place.The goal is to be able to at least touch upon so-called Lubin-Tate formal group laws, which lead to an explicit construction of the abelian extensions of a local field. We will follow the first few sections of Riehl's senior thesis and some parts of Milne's notes (see below). Topics for further individual study in the latter stages of the project include: 1) computing several explicit examples; 2) introducing and discussing ramification groups; 2) discussing some of the main theorems about local fields (e.g. local reciprocity, local Kronecker-Weber etc.) PrerequisitesGalois Theory III.Resources |
email: Alexander Stasinski