Complete field

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

Real and complex numbers

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

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

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

  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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