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 \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""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 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)).