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

  $$|\alpha |=\sqrt[n]{|N(\alpha )|}\mathit{\hspace{1em}}(\alpha \in K),$$

  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