extension of Krull valuation

The Krull valuation||K  of the field K is called the of the Krull valuation||k  of the field k, if k is a subfieldMathworldPlanetmath of K and  ||k  is the restriction of  ||K  on k.

Theorem 1.

The trivial valuation is the only of the trivial valuation of k to an algebraic extensionMathworldPlanetmath field K of k.

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


where  a1,,ank.  Since  |aj|1  for all j’s, we get the impossibility


(cf. the sharpening of the ultrametric triangle inequality).  Therefore we must have  |ξ|1  for all  ξK,  and because the condition  0<|ξ|<1  would imply that  |ξ-1|>1,  we see that


which that the valuationMathworldPlanetmath 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.


[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