Complete field explained

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

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^[1] 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 = P ^{1} at p = [ 0 : 1 ], the origin in the affine chart x _{1} ≠ 0 , then the completion of k(X) at p is isomorphic to the power-series ring k((x)).

Notes and References

1. Book: Koblitz, Neal.. P-adic Numbers, p-adic Analysis, and Zeta-Functions. 1984. Springer New York. 978-1-4612-1112-9. Second. New York, NY. 52–75. 853269675.