### Abstract

There seem few examples in the literature of quartic surfaces defined over ℚ that are everywhere locally solvable, yet which have no global point. It is a delicate question as to whether such surfaces can possess points defined over an odd-degree number field, and to our knowledge no previous example is known. We give here an example of such a diagonal quartic surface which contains a point defined over a cubic extension field (and it follows that there exist number fields of every odd degree greater than 1 in which the surface has points). This surface is one member of a more general family of surfaces, each of which is also everywhere locally solvable but with no rational point.

Original language | English (US) |
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Pages (from-to) | 253-260 |

Number of pages | 8 |

Journal | Publicationes Mathematicae |

Volume | 93 |

Issue number | 1-2 |

DOIs | |

State | Published - Jan 1 2018 |

### Keywords

- Hilbert symbol
- Quartic surface

### ASJC Scopus subject areas

- Mathematics(all)

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## Cite this

*Publicationes Mathematicae*,

*93*(1-2), 253-260. https://doi.org/10.5486/PMD.2018.8252