Let k be a global field and \pp any nonarchimedean prime of k. We give a newand uniform proof of the well known fact that the set of all elements of kwhich are integral at \pp is diophantine over k. Let k^{perf} be the perfectclosure of a global field of characteristic p>2. We also prove that the set ofall elements of k^{perf} which are integral at some prime \qq of k^{perf} isdiophantine over k^{perf}, and this is the first such result for a field whichis not finitely generated over its constant field. This is related to Hilbert'sTenth Problem because for global fields k of positive characteristic, giving adiophantine definition of the set of elements that are integral at a prime isone of two steps needed to prove that Hilbert's Tenth Problem for k isundecidable.
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