Complete field

Last updated September 07, 2024

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 $\mathbb {C}$ (since its absolute Galois group is $\mathbb {Z} /2$ ). In this case, $\mathbb {C}$ 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^[1] is not algebraically closed. Typically, the process is to take the separable closure and then complete it again. This field is usually denoted $\mathbb {C} _{p}.$

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 $x_{1}\neq 0,$ then the completion of $k(X)$ at $p$ is isomorphic to the power-series ring $k((x)).$

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References

↑ 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.