  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.
