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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

### Real and complex numbers

The real numbers are the field with the standard euclidean metric ${\displaystyle |x-y|}$ . Since it is constructed from the completion of ${\displaystyle \mathbb {Q} }$  with respect to this metric, it is a complete field. Extending the reals by its algebraic closure gives the field ${\displaystyle \mathbb {C} }$  (since its absolute Galois group is ${\displaystyle \mathbb {Z} /2}$ ). In this case, ${\displaystyle \mathbb {C} }$  is also a complete field, but this is not the case in many cases.

The p-adic numbers are constructed from ${\displaystyle \mathbb {Q} }$  by using the p-adic absolute value

${\displaystyle v_{p}(a/b)=v_{p}(a)-v_{p}(b)}$

where ${\displaystyle a,b\in \mathbb {Z} .}$  Then using the factorization ${\displaystyle a=p^{n}c}$  where ${\displaystyle p}$  does not divide ${\displaystyle c,}$  its valuation is the integer ${\displaystyle n}$ . The completion of ${\displaystyle \mathbb {Q} }$  by ${\displaystyle v_{p}}$  is the complete field ${\displaystyle \mathbb {Q} _{p}}$  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 ${\displaystyle \mathbb {C} _{p}.}$

### Function field of a curve

For the function field ${\displaystyle k(X)}$  of a curve ${\displaystyle X/k,}$  every point ${\displaystyle p\in X}$  corresponds to an absolute value, or place, ${\displaystyle v_{p}}$ . Given an element ${\displaystyle f\in k(X)}$  expressed by a fraction ${\displaystyle g/h,}$  the place ${\displaystyle v_{p}}$  measures the order of vanishing of ${\displaystyle g}$  at ${\displaystyle p}$  minus the order of vanishing of ${\displaystyle h}$  at ${\displaystyle p.}$  Then, the completion of ${\displaystyle k(X)}$  at ${\displaystyle p}$  gives a new field. For example, if ${\displaystyle X=\mathbb {P} ^{1}}$  at ${\displaystyle p=[0:1],}$  the origin in the affine chart ${\displaystyle x_{1}\neq 0,}$  then the completion of ${\displaystyle k(X)}$  at ${\displaystyle p}$  is isomorphic to the power-series ring ${\displaystyle k((x)).}$

## 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.