Complete field

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.

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 ${\displaystyle \mathbb {C} _{p}}$.

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