TY - JOUR

T1 - Integrality at a prime for global fields and the perfect closure of global fields of characteristic p > 2

AU - Eisentraeger, Anne Kirsten

PY - 2005/9/1

Y1 - 2005/9/1

N2 - 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 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 q 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 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.

AB - 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 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 q 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 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.

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U2 - 10.1016/j.jnt.2005.02.008

DO - 10.1016/j.jnt.2005.02.008

M3 - Article

AN - SCOPUS:23844487393

VL - 114

SP - 170

EP - 181

JO - Journal of Number Theory

JF - Journal of Number Theory

SN - 0022-314X

IS - 1

ER -