# extension of Krull valuation

The Krull valuation $|\cdot {|}_{K}$ of the field $K$ is called the of the Krull valuation $|\cdot {|}_{k}$ of the field $k$, if $k$ is a subfield^{} of $K$ and $|\cdot {|}_{k}$ is the restriction of $|\cdot {|}_{K}$ on $k$.

###### Theorem 1.

The trivial valuation is the only of the trivial valuation of $k$ to an algebraic extension^{} field $K$ of $k$.

Proof. Let’s denote by $|\cdot |$ the trivial valuation of $k$ and also its arbitrary Krull to $K$. Suppose that there is an element $\alpha $ of $K$ such that $|\alpha |>1$. This element satisfies an algebraic equation

$${\alpha}^{n}+{a}_{1}{\alpha}^{n-1}+\mathrm{\dots}+{a}_{n}=0,$$ |

where ${a}_{1},\mathrm{\dots},{a}_{n}\in k$. Since $|{a}_{j}|\leqq 1$ for all $j$’s, we get the impossibility

$$0=|{\alpha}^{n}+{a}_{1}{\alpha}^{n-1}+\mathrm{\dots}+{a}_{n}|=\mathrm{max}\{{|\alpha |}^{n},|{a}_{1}|\cdot {|\alpha |}^{n-1},\mathrm{\dots},|{a}_{n}|\}={|\alpha |}^{n}>1$$ |

(cf. the sharpening of the ultrametric triangle inequality). Therefore we must have $|\xi |\leqq 1$ for all $\xi \in K$, and because the condition $$ would imply that $|{\xi}^{-1}|>1$, we see that

$$|\xi |=1\mathit{\hspace{1em}}\forall \xi \in K\setminus \{0\},$$ |

which that the valuation^{} is trivial.

The proof (in [1]) of the next “extension theorem” is much longer (one must utilize the extension theorem concerning the place of field):

###### Theorem 2.

Every Krull valuation of a field $k$ can be extended to a Krull valuation of any field of $k$.

## References

[1] Emil Artin: . Lecture notes. Mathematisches Institut, Göttingen (1959).

Title | extension of Krull valuation |
---|---|

Canonical name | ExtensionOfKrullValuation |

Date of creation | 2013-03-22 14:55:57 |

Last modified on | 2013-03-22 14:55:57 |

Owner | pahio (2872) |

Last modified by | pahio (2872) |

Numerical id | 13 |

Author | pahio (2872) |

Entry type | Theorem |

Classification | msc 13F30 |

Classification | msc 13A18 |

Classification | msc 12J20 |

Classification | msc 11R99 |

Related topic | GelfandTornheimTheorem |