Local Field

In mathematics, a local field is a special type of field that is a locally compact topological field with respect to a non-discrete topology. Given such a field, an absolute value can be defined on it. There are two basic types of local field: those in which the absolute value is archimedean and those in which it is not. In the first case, one calls the local field an archimedean local field, in the second case, one calls it a non-archimedean local field. Local fields arise naturally in number theory as completions of global fields.

Every local field is isomorphic (as a topological field) to one of the following:

• Archimedean local fields (characteristic zero): the real numbers R, and the complex numbers C.
• Non-archimedean local fields of characteristic zero: finite extensions of the p-adic numbers Q_p (where p is any prime number).
• Non-archimedean local fields of characteristic p (for p any given prime number): the field of formal Laurent series F_q((T)) over a finite field F_q (where q is a power of p).

There is an equivalent definition of non-archimedean local field: it is a field that is complete with respect to a discrete valuation and whose residue field is finite. However, some authors consider a more general notion, requiring only that the residue field be perfect, not necessarily finite. This article uses the former definition.