Local Fields

Alternate Title

Local Fields

Document Type

Presentation

Publication Date

2024

Published In

Preliminary Arizona Winter School 2024: Symmetries Of Root Systems And Local Fields

Abstract

In 1897, Kurt Hensel introduced the p-adic numbers as a way to apply techniques involving power series within the context of number theory. The p-adic numbers are an example of a local field---that is, a field arising as a suitable completion of either a number field or a function field over a finite field. In modern number theory, many deep questions about the arithmetic of number fields have been approached by first investigating their local versions, with applications ranging from Diophantine geometry to the Langlands program.

Starting from discrete valuations, this course will explore the theory of local fields. The first half of the course will focus on arithmetic properties of local fields, highlighting applications of Hensel's lemma to quadratic forms. In the second half of the course, we will shift to the Galois theory of local fields, with an emphasis on ramification groups. Time permitting, we will conclude the course with an application of local Galois theory to the proof of the Kronecker--Weber Theorem.

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