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

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2 Citations (Scopus)

Abstract

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.

Original languageEnglish (US)
Pages (from-to)170-181
Number of pages12
JournalJournal of Number Theory
Volume114
Issue number1
DOIs
StatePublished - Sep 1 2005

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Integrality
Closure
Hilbert's Tenth Problem
Positive Characteristic
Finitely Generated

All Science Journal Classification (ASJC) codes

  • Algebra and Number Theory

Cite this

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abstract = "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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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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