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 $\Q$ with respect to this metric, it is a complete field. Extending the reals by its algebraic closure gives the field $\Complex$ (since its absolute Galois group is $\Z/2$). In this case, $\Complex$ is also a complete field, but this is not the case in many cases.

p-adic

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

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

where $a,b \in \Z.$ Then using the factorization $a = p^nc$ where $p$ does not divide $c,$ its valuation is the integer $n$. The completion of $\Q$ by $v_p$ is the complete field $\Q_p$ 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 $\Complex_p.$

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 $X = \mathbb^1$ at p = :1 the origin in the affine chart $x_1 \neq 0,$ then the completion of $k(X)$ at $p$ is isomorphic to the power-series ring $k((x)).$

References

See also

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Field (mathematics)

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