## 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) |
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Pages (from-to) | 170-181 |

Number of pages | 12 |

Journal | Journal of Number Theory |

Volume | 114 |

Issue number | 1 |

DOIs | |

State | Published - Sep 2005 |

## All Science Journal Classification (ASJC) codes

- Algebra and Number Theory