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

p-adic

The p-adic numbers are constructed from

$\mathbb{Q}$

by using the p-adic absolute value

$v_p(a/b)=v_p(a)-v_p(b)$

where

$a,b\in \mathbb{Z}.$

Then using the factorization

$a=p^{n}c$

where

$p$

does not divide

$c,$

its valuation is the integer

$n$

. The completion of

$\mathbb{Q}$

by

$v_p$

is the complete field

${\mathbb{Q}}_p$

called the p-adic numbers. This is a case where the field is not algebraically closed. Typically, the process is to take the separable closure and then complete it again. This field is usually denoted

${ℂ}_p.$

Function field of a curve

For the function field $k(X)$ of a curve $X/k,$ every point $p\in X$ corresponds to an absolute value, or place, $v_p$. Given an element $f\in k(X)$ expressed by a fraction $g/h,$ the place $v_p$ measures the order of vanishing of $g$ at $p$ minus the order of vanishing of $h$ at $p.$ Then, the completion of $k(X)$ at $p$ gives a new field. For example, if $X={\mathbb{P}}^1$ at $p=[0:1],$ the origin in the affine chart $x_1\ne 0,$ then the completion of $k(X)$ at $p$ is isomorphic to the power-series ring $k((x)).$

See also

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