# value group of completion

Let $k$ be a field and $|\cdot |$ its non-archimedean valuation of rank one (http://planetmath.org/KrullValuation). Then its value group $|k\setminus \{0\}|$ may be considered to be a subgroup^{} of the multiplicative group^{} of $\mathbb{R}$. In the completion $K$ of the valued field $k$, the extension^{} of the valuation^{} is defined by

$$|x|=:\underset{n\to \mathrm{\infty}}{lim}|{x}_{n}|,$$ |

when the Cauchy sequence^{} ${x}_{1},{x}_{2},\mathrm{\dots},{x}_{n},\mathrm{\dots}$ of elements of $k$ determines the element $x$ of $K$.

###### Theorem.

The non-archimedean field $k$ and its completion $K$ have the same value group.

Proof. Of course, $|k|\subseteq |K|$. Let $x={lim}_{n\to \mathrm{\infty}}{x}_{n}$ be any non-zero element of $K$, where ${x}_{j}$’s form a Cauchy sequence in $k$. Then there exists a positive number ${n}_{0}$ such that

$$ |

for all $n>{n}_{0}$. For all these values of $n$ we have

$$|{x}_{n}|=|x+({x}_{n}-x)|=|x|$$ |

according to the ultrametric triangle inequality. Thus we see that $|K|\subseteq |k|$.

Title | value group of completion |

Canonical name | ValueGroupOfCompletion |

Date of creation | 2013-03-22 14:58:14 |

Last modified on | 2013-03-22 14:58:14 |

Owner | pahio (2872) |

Last modified by | pahio (2872) |

Numerical id | 10 |

Author | pahio (2872) |

Entry type | Theorem |

Classification | msc 13F30 |

Classification | msc 13J10 |

Classification | msc 13A18 |

Classification | msc 12J20 |

Related topic | KrullValuation |

Related topic | ExtensionOfValuationFromCompleteBaseField |

Defines | value group of the completion |