Uniqueness of topology for the 𝑝-adic integers

Abstract

It is shown that the only Hausdorff topologies on Z p {{\mathbf {Z}}_p} , the p p -adic integers, which make it into a locally compact Abelian group are the p p -adic and discrete topologies. The key ingredient in the proof is a structure theorem for certain LCA groups which may be of independent interest.