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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 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 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.
Keywords:
algebraic field extension
Related:
CompleteUltrametricField, ValueGroupOfCompletion, NthRoot
Type of Math Object:
Theorem
Major Section:
Reference
Parent:
Groups audience:
Mathematics Subject Classification
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