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

Abstract

Let k be a global field and \pp 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 \pp 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 \qq 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.