In arithmetic geometry, people study systems of polynomial equations and their solutions over number fields and other non-algebraically closed domains. Even the problem of deciding whether there are any solutions at all over such domains is, in general, an unsolved problem.

Interestingly enough, the question becomes easier if one enlarges the domain. For instance, if one puts a topology on a number field and takes the completion, then any solvable system of equations over the number field necessarily also has a solution over this larger field. The latter turns out to be a solvable problem. I will explain how one can do this in practice.