local field
A local field is a topological field which is Hausdorff^{} and locally compact as a topological space^{}.
Examples of local fields include:

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Any field together with the discrete topology.

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The field $\mathbb{R}$ of real numbers.

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The field $\u2102$ of complex numbers^{}.

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The field ${\mathbb{Q}}_{p}$ of $p$–adic rationals (http://planetmath.org/PAdicIntegers), or any finite extension^{} thereof.

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The field ${\mathbb{F}}_{q}((t))$ of formal Laurent series in one variable $t$ with coefficients in the finite field^{} ${\mathbb{F}}_{q}$ of $q$ elements.
In fact, this list is complete^{}—every local field is isomorphic as a topological field to one of the above fields.
1 Acknowledgements
This document is dedicated to those who made it all the way through Serre’s book [1] before realizing that nowhere within the book is there a definition of the term “local field.”
References
 1 Jean–Pierre Serre, Local Fields, Springer–Verlag, 1979 (GTM 67).
Title  local field 

Canonical name  LocalField 
Date of creation  20130322 12:48:07 
Last modified on  20130322 12:48:07 
Owner  djao (24) 
Last modified by  djao (24) 
Numerical id  7 
Author  djao (24) 
Entry type  Definition 
Classification  msc 13H99 
Classification  msc 12J99 
Classification  msc 11S99 