## Abstract

We prove that a quintic form in 26 variables defined over a p-adic field K always has a non trivial zero over K if the residue class field of K has at least 47 elements. This is in agreement with the theorem of Ax-Kochen which states that a homogeneous form of degree d in d^{2} + 1 variables defined over Q_{p} has a nontrivial Q_{p}-rational zero if p is sufficiently large. The Ax-Kochen theorem gives no results on the bound for p. For d= 1, 2, 3 it has been known for a long time that there is a nontrivial Q_{p}-rational zero for all values of p. For d=4, Terjanian gave an example of a form in 18 variables over Q_{2} having no nontrivial Q_{2}-rational zero. This is the first result which gives an effective bound for the case d= 5.

Original language | English |
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Pages (from-to) | 231-241 |

Number of pages | 11 |

Journal | Journal of Number Theory |

Volume | 57 |

Issue number | 2 |

DOIs | |

State | Published - Apr 1996 |

## ASJC Scopus subject areas

- Algebra and Number Theory