In mathematics, local class field theory (LCFT), introduced by Helmut Hasse, is the study of abelian extensions of local fields; here, "local field" means a field which is complete with respect to an absolute value or a discrete valuation with a finite residue field: hence every local field is isomorphic (as a topological field) to the real numbers R, the complex numbers C, a finite extension of the p-adic numbers Qp (where p is any prime number), or the field of formal Laurent series Fq((T)) over a finite field Fq.
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In this Dossier
- 👉 Discrete valuation in the context of Local class field theory
- Discrete valuation in the context of Local field
Discrete valuation in the context of Local field
In mathematics, a local field is a certain type of topological field: by definition, a local field is a locally compact Hausdorff non-discrete topological field. Local fields find many applications in algebraic number theory, where they arise naturally as completions of global fields. Further, tools like integration and Fourier analysis are available for functions defined on local fields.
Given a local field, an absolute value can be defined on it which gives rise to a complete metric that generates its topology. There are two basic types of local field: those called Archimedean local fields in which the absolute value is Archimedean, and those called non-Archimedean local fields in which it is not. The non-Archimedean local fields can also be defined as those fields which are complete with respect to a metric induced by a discrete valuation v whose residue field is finite.
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