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Definition. Let K be a field. An absolute value on K is a function | · | : K → R such that:
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. Equivalently, a local field is a locally compact topological field with respect to a non-discrete topology.
The goal of this book is to present local class field theory from the cohomo logical point of view, following the method inaugurated by Hochschild and developed by Artin-Tate. This theory is about extensions-primarily abelian-of "local" (i.e., complete for a discrete valuation) fields with finite residue field.
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20 de dic. de 2014 · A field that is complete with respect to a discrete valuation and has finite residue field. The structure of a local field $K$ is well known: 1) if the characteristic of $K$ is $0$, then $K$ is a finite extension of the field $\mathbb{Q}_p$ of $p$-adic numbers; 2) if the characteristic of $K$ is greater than $0$, then $K$ is isomorphic to the ...
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