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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.〔Page 20 of 〕 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): finite extensions of 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.〔See, for example, definition 1.4.6 of 〕 This article uses the former definition. ==Induced absolute value== Given a locally compact topological field ''K'', an absolute value can be defined as follows. First, consider the additive group of the field. As a locally compact topological group, it has a unique (up to positive scalar multiple) Haar measure μ. The absolute value is defined so as to measure the change in size of a set after multiplying it by an element of ''K''. Specifically, define |·| : ''K'' → R by〔Page 4 of 〕 : for any measurable subset ''X'' of ''K'' (with 0 < μ(X) < ∞). This absolute value does not depend on ''X'' nor on the choice of Haar measure (since the same scalar multiple ambiguity will occur in both the numerator and the denominator). This definition is very similar to that of the modular function. Given such an absolute value on ''K'', a new induced topology can be defined on ''K''. This topology is the same as the original topology.〔Corollary 1, page 5 of 〕 Explicitly, for a positive real number ''m'', define the subset ''B''m of ''K'' by : Then, the ''B''m make up a neighbourhood basis of 0 in ''K''. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Local field」の詳細全文を読む スポンサード リンク
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