# Corps valu\'es locaux

## TL;DR

This paper explores local fields, such as p-adic numbers and formal Laurent series, highlighting their foundational role in number theory, algebra, and topology, and discussing their applications in elementary and algebraic number theory.

## Contribution

It provides an overview of local fields, their properties, and applications, connecting general algebraic and topological concepts to specific local field examples.

## Key findings

- Local fields bridge number theory, algebra, and topology.
- p-adic numbers and Laurent series are key examples.
- Applications in elementary and algebraic number theory.

## Abstract

Many active mathematical research topics nowadays include the concepts of valued fields and local fields, especially the local field of p-adic numbers Qp and the field of formal Laurent series F((X)). Local fields are a notion situated in the boundary between number theory, algebra and topology. They use many definitions and theorems - more or less advanced - of general algebra and topology. Gradually, we will go from the general to the local, from the valued fields to the local fields, of which we will discuss some applications, especially in elementary and algebraic number theory.