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{{Cite book|last=Koblitz, Neal.|url=https://www.worldcat.org/oclc/853269675|title=P-adic Numbers, p-adic Analysis, and Zeta-Functions|date=1984|publisher=Springer New York|isbn=978-1-4612-1112-9|edition= Second|location=New York, NY|pages=52–75|oclc=853269675}} 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 $X = \mathbb{P}^1$ at $p = [0:1],$ 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 $k((x)).$

complete field

Constructions

=Real and complex numbers=

=p-adic=

=Function field of a curve=

References

See also