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 $\mathbb {C}$ (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 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 $k((x)).$

See also

• Completion (algebra) – in algebra, any of several related functors on rings and modules that result in complete topological rings and modulesPages displaying wikidata descriptions as a fallback
• Complete topological vector space – A TVS where points that get progressively closer to each other will always converge to a point
• Hensel's lemma – Result in modular arithmetic
• Henselian ring – local ring in which Hensel’s lemma holdsPages displaying wikidata descriptions as a fallback
• Compact group – Topological group with compact topology
• Locally compact field
• Locally compact quantum group – relatively new C*-algebraic approach toward quantum groupsPages displaying wikidata descriptions as a fallback
• Locally compact group – topological group for which the underlying topology is locally compact and Hausdorff, so that the Haar measure can be definedPages displaying wikidata descriptions as a fallback
• Ordered topological vector space
• Ostrowski's theorem – On all absolute values of rational numbers
• Topological abelian group – topological group whose group is abelianPages displaying wikidata descriptions as a fallback
• Topological field – Algebraic structure with addition, multiplication, and divisionPages displaying short descriptions of redirect targets
• Topological group – Group that is a topological space with continuous group action
• Topological module
• Topological ring – ring where ring operations are continuousPages displaying wikidata descriptions as a fallback
• Topological semigroup – semigroup with continuous operationPages displaying wikidata descriptions as a fallback
• Topological vector space – Vector space with a notion of nearness