Local 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.^[1] Equivalently, a local field is a locally compact topological field with respect to a non-discrete topology.^[2] Sometimes, the real numbers R, and the complex numbers C (with their standard topologies) are also defined to be local fields; this is the convention we will adopt below. 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.^[3] Local fields arise naturally in number theory as completions of global fields.^[4]

While Archimedean local fields have been quite well known in mathematics for at least 250 years, the first examples of non-Archimedean local fields, the fields of p-adic numbers for positive prime integer p, were introduced by Kurt Hensel at the end of the 19th century.

Every local field is isomorphic (as a topological field) to one of the following:^[3]

• 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.

In particular, of importance in number theory, classes of local fields show up as the completions of algebraic number fields with respect to their discrete valuation corresponding to one of their maximal ideals. Research papers in modern number theory often consider a more general notion, requiring only that the residue field be perfect of positive characteristic, not necessarily finite.^[5] This article uses the former definition.