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Hagen
Posts:
438
Registered:
12/13/04
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Re: Non-Archimedean Field Valuation Property
Posted:
Jan 12, 2006 6:48 AM
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> What is meant by valuation in that site is the
> definition of absolute value. Anyhow I found this
> approach in the book "Algebraic extensions of fields
> " by Paul McCarthy .
> I think it's much easier to study in this way the
> analytic properties of the respective fields. On the
> other hand, I think this approach is outdated.
No, the approach is not outdated.
Absolute values on fields and real valuations are different things: An absolute value that does not satisfy the ultrametric triangle imequality does not yield a valuation.
This fact leads to the (sometimes unsatisfactory) situation that one has to develop two theories (one for absolute values, one for valuations) that coincide in the rather large area of real valuations. However there are important differences. To mention one: an absolute value on a field K in general can not be extended to a field extension L of K. A valuation on the other hand can always be extended.
H
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