# extension of valuation from complete base field

Here the valuations are of rank one, and it may be supposed that the values are real numbers.

• Assume a finite field extension $K/k$ and a valuation of $K$.  If the base field is complete (http://planetmath.org/Complete) with regard to this valuation, so is also the extension field.

• If $K/k$ is an algebraic field extension and if the base field $k$ is complete (http://planetmath.org/Complete) with regard to its valuation  $|\cdot|$,   then this valuation has one and only one extension to the field $K$.  This extension is determined by

 $|\alpha|=\sqrt[n]{|N(\alpha)|}\quad(\alpha\in K),$

where $N(\alpha)$ is the norm of the element $\alpha$ in the simple field extension $k(\alpha)/k$ and $n$ is the degree of this field extension.

These theorems concern also Archimedean valuations.

 Title extension of valuation from complete base field Canonical name ExtensionOfValuationFromCompleteBaseField Date of creation 2013-03-22 15:01:01 Last modified on 2013-03-22 15:01:01 Owner pahio (2872) Last modified by pahio (2872) Numerical id 9 Author pahio (2872) Entry type Theorem Classification msc 13F30 Classification msc 13A18 Classification msc 12J20 Classification msc 11R99 Related topic CompleteUltrametricField Related topic ValueGroupOfCompletion Related topic NthRoot