# complete ultrametric field

A field $K$ equipped with a non-archimedean valuation $|\cdot |$ is called a non-archimedean field or also an ultrametric field, since the valuation^{} the ultrametric $d(x,y):=|x-y|$ of $K$.

###### Theorem.

Let $(K,d)$ be a complete^{} (http://planetmath.org/Complete) ultrametric field. A necessary and sufficient condition for the convergence of the series

${a}_{1}+{a}_{2}+{a}_{3}+\mathrm{\dots}$ | (1) |

in $K$ is that

$\underset{n\to \mathrm{\infty}}{lim}{a}_{n}=\mathrm{\hspace{0.33em}0}.$ | (2) |

Proof. Let $\epsilon $ be any positive number. When (1) converges^{}, it satisfies the Cauchy condition and therefore exists a number ${m}_{\epsilon}$ such that surely

$$ |

for all $m\geqq {m}_{\epsilon}$; thus (2) is necessary. On the contrary, suppose the validity of (2). Now one may determine such a great number ${n}_{\epsilon}$ that

$$ |

No matter how great is the natural number^{} $n$, the ultrametric then guarantees the inequality

$$ |

always when $m\geqq {n}_{\epsilon}$. Thus the partial sums of (1) form a Cauchy sequence^{}, which converges in the complete field. Hence the series (1) converges, and (2) is sufficient.

Title | complete ultrametric field |

Canonical name | CompleteUltrametricField |

Date of creation | 2013-03-22 14:55:37 |

Last modified on | 2013-03-22 14:55:37 |

Owner | pahio (2872) |

Last modified by | pahio (2872) |

Numerical id | 15 |

Author | pahio (2872) |

Entry type | Theorem |

Classification | msc 12J10 |

Classification | msc 54E35 |

Related topic | Series |

Related topic | NecessaryConditionOfConvergence |

Related topic | ExtensionOfValuationFromCompleteBaseField |

Related topic | PropertiesOfNonArchimedeanValuations |

Defines | ultrametric field |

Defines | non-archimedean field |