extension of Krull valuation
Proof. Let’s denote by the trivial valuation of and also its arbitrary Krull to . Suppose that there is an element of such that . This element satisfies an algebraic equation
where . Since for all ’s, we get the impossibility
(cf. the sharpening of the ultrametric triangle inequality). Therefore we must have for all , and because the condition would imply that , we see that
which that the valuation is trivial.
Every Krull valuation of a field can be extended to a Krull valuation of any field of .
 Emil Artin: . Lecture notes. Mathematisches Institut, Göttingen (1959).
|Title||extension of Krull valuation|
|Date of creation||2013-03-22 14:55:57|
|Last modified on||2013-03-22 14:55:57|
|Last modified by||pahio (2872)|