Integrality at a prime for global fields and the perfect closure of global fields of characteristic p > 2

Let k be a global field and p any nonarchimedean prime of k. We give a new and uniform proof of the well known fact that the set of all elements of k which are integral at p is diophantine over k. Let k^perf be the perfect closure of a global field of characteristic p > 2. We also prove that the set of all elements of k^perf which are integral at some prime q of k^perf is diophantine over k^perf, and this is the first such result for a field which is not finitely generated over its constant field. This is related to Hilbert's Tenth Problem because for global fields k of positive characteristic, giving a diophantine definition of the set of elements that are integral at a prime is one of two steps needed to prove that Hilbert's Tenth Problem for k is undecidable.