Set Theory and p-adic Algebras

TL;DR

This paper explores the independence of certain maximal ideals in p-adic algebras from ZFC and establishes a connection between P-points in a universal totally disconnected compactification and p-adic analysis, contrasting it with Stone-Cech theory.

Contribution

It introduces a novel approach using p-adic analysis to relate P-points in a universal totally disconnected compactification to maximal ideals, extending the theory beyond Stone-Cech methods.

Findings

Existence of a maximal ideal of height 0 is independent of ZFC.

Established a relation between P-points in the boundary and p-adic algebra.

Contrasted the lifting properties of Stone-Cech and universal totally disconnected compactifications.

Abstract

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.