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

Let k be a global field and image 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 image is diophantine over k. Let kperf be the perfect closure of a global field of characteristic p>2. We also prove that the set of all elements of kperf which are integral at some prime image of kperf is diophantine over kperf, and this is the first such result for a field which is not finitely generated over its constant field. This is related to Hilbertʹs 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 Hilbertʹs Tenth Problem for k is undecidable.