### 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 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 q 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.

Original language | English (US) |
---|---|

Pages (from-to) | 170-181 |

Number of pages | 12 |

Journal | Journal of Number Theory |

Volume | 114 |

Issue number | 1 |

DOIs | |

State | Published - Sep 1 2005 |

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### All Science Journal Classification (ASJC) codes

- Algebra and Number Theory

### Cite this

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**Integrality at a prime for global fields and the perfect closure of global fields of characteristic p > 2.** / Eisentraeger, Anne Kirsten.

Research output: Contribution to journal › Article

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.

UR - http://www.scopus.com/inward/record.url?scp=23844487393&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=23844487393&partnerID=8YFLogxK

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 -