Local field

Locally compact topological field

In mathematics, a field K is called a non-Archimedean local field if it is complete with respect to a metric induced by a discrete valuation v and if its residue field k is finite. In general, a local field is a locally compact topological field with respect to a non-discrete topology. The real numbers R, and the complex numbers C are Archimedean local fields. Given a local field, the valuation defined on it can be of either of two types, each one corresponds to one of the two basic types of local fields: those in which the valuation 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.

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