Completion of a valued field

Summary

The completion of a field with an absolute value is a larger such field, see the definition. In it every Cauchy sequence converges, and the original field is dense. It is the completion of the metric space of the field, with the field operations extended by continuity. The real numbers are the completion of $\Q$ for the usual absolute value, and the p-adic numbers are its completion for the p-adic absolute value. Many problems over a valued field are solved first in its completion, where limits exist and Hensel's lemma applies.