Higher Dimensional Local Fields
It is natural to introduce non-archimedean local fields in a uniform geometric way as the field of fractions of the completion of the local ring of a one-dimensional arithmetic scheme of rank 1 at its non-singular point. For generalizations, a local field is sometimes called a one-dimensional local field.
For a non-negative integer n, an n-dimensional local field is a complete discrete valuation field whose residue field is an (n − 1)-dimensional local field. Depending on the definition of local field, a zero-dimensional local field is then either a finite field (with the definition used in this article), or a quasi-finite field, or a perfect field.
From the geometric point of view, n-dimensional local fields with last finite residue field are naturally associated to a complete flag of subschemes of an n-dimensional arithmetic scheme.
Read more about this topic: Local Field
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