  We verified that the existence of a maximal ideal of height 0 in a p-adic
algebra in a certain class is independent of the axiom of ZFC. We established
the theory on a P-point in the boundary of a topological space in the universal
totally disconnected Hausdorff compactification. It is quite similar with the
theory on a P-point in the boundary of a topological space in the Stone-Cech
compactification. The latter theory relies on the real analysis, and the reason
why the real analysis works for it is because the Stone-Cech compactification
has the lifting property for a real bounded continuous function. On the other
hand, the universal totally disconnected Hausdorff compactification does not
have the lifting property for a real bounded continuous function in general,
and hence the same technique with the real analysis is not valid for the former
theory. We applied the p-adic analysis instead, and it yields a relation with a
P-point in the boundary in the universal totally disconnected Hausdorff
compactification and a maximal ideal of height 0 in the corresponding p-adic
algebra.
