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).

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 $$\Q$$ with respect to this metric, it is a complete field. Extending the reals by its algebraic closure gives the field $$\Complex$$ (since its absolute Galois group is $$\Z/2$$). In this case, $$\Complex$$ is also a complete field, but this is not the case in many cases.

p-adic
The p-adic numbers are constructed from $$\Q$$ by using the p-adic absolute value"$v_p(a/b) = v_p(a) - v_p(b)$"where $$a,b \in \Z.$$ Then using the factorization $$a = p^nc$$ where $$p$$ does not divide $$c,$$ its valuation is the integer $$n$$. The completion of $$\Q$$ by $$v_p$$ is the complete field $$\Q_p$$ 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 $$\Complex_p.$$

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 $$k((x)).$$