In mathematics, a higher (-dimensional) local field is an important example of a complete discrete valuation field. Such fields are also sometimes called multi-dimensional local fields.
On the usual local fields (typically completions of number fields or the quotient fields of local rings of algebraic curves) there is a unique surjective discrete valuation (of rank 1) associated to a choice of a local parameter of the fields, unless they are archimedean local fields such as the real numbers and complex numbers. Similarly, there is a discrete valuation of rank n on almost all n-dimensional local fields, associated to a choice of n local parameters of the field.[1] In contrast to one-dimensional local fields, higher local fields have a sequence of residue fields.[2] There are different integral structures on higher local fields, depending how many residue fields information one wants to take into account.[2]
Geometrically, higher local fields appear via a process of localization and completion of local rings of higher dimensional schemes.[2] Higher local fields are an important part of the subject of higher dimensional number theory, forming the appropriate collection of objects for local considerations.
^Fesenko, I.B., Vostokov, S.V. Local Fields and Their Extensions. American Mathematical Society, 1992, Chapter 1 and Appendix.
^ abcFesenko, I., Kurihara, M. (eds.) Invitation to Higher Local Fields. Geometry and Topology Monographs, 2000, section 1 (Zhukov).
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