Complete field

Summary

In mathematics, a complete field is a field equipped with a metric and complete with respect to that metric. Basic examples include the real numbers, the complex numbers, and complete valued fields (such as the p-adic numbers).

Constructions

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Real and complex numbers

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The real numbers are the field with the standard euclidean metric  . Since it is constructed from the completion of   with respect to this metric, it is a complete field. Extending the reals by its algebraic closure gives the field   (since its absolute Galois group is  ). In this case,   is also a complete field, but this is not the case in many cases.

p-adic

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The p-adic numbers are constructed from   by using the p-adic absolute value

 

where   Then using the factorization   where   does not divide   its valuation is the integer  . The completion of   by   is the complete field   called the p-adic numbers. This is a case where the field[1] is not algebraically closed. Typically, the process is to take the separable closure and then complete it again. This field is usually denoted  

Function field of a curve

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For the function field   of a curve   every point   corresponds to an absolute value, or place,  . Given an element   expressed by a fraction   the place   measures the order of vanishing of   at   minus the order of vanishing of   at   Then, the completion of   at   gives a new field. For example, if   at   the origin in the affine chart   then the completion of   at   is isomorphic to the power-series ring  

References

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  1. ^ Koblitz, Neal. (1984). P-adic Numbers, p-adic Analysis, and Zeta-Functions (Second ed.). New York, NY: Springer New York. pp. 52–75. ISBN 978-1-4612-1112-9. OCLC 853269675.

See also

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