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 ${\displaystyle k((x)).}$