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

p-adic

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

${\displaystyle v_{p}(a/b)=v_{p}(a)-v_{p}(b)}$

where ${\displaystyle a,b\in \mathbb {Z} .}$ Then using the factorization ${\displaystyle a=p^{n}c}$ where ${\displaystyle p}$ does not divide ${\displaystyle c,}$ its valuation is the integer ${\displaystyle n}$. The completion of ${\displaystyle \mathbb {Q} }$ by ${\displaystyle v_{p}}$ is the complete field ${\displaystyle \mathbb {Q} _{p}}$ 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 ${\displaystyle \mathbb {C} _{p}.}$

Function field of a curve

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

References

1. 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.

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 G 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  – concept in mathematicsPages 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