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NOTES ON EXTREMAL AND TAME VALUED FIELDS

Published online by Cambridge University Press:  29 June 2016

SYLVY ANSCOMBE  and
FRANZ-VIKTOR KUHLMANN
SYLVY ANSCOMBE
Affiliation:
JEREMIAH HORROCKS INSTITUTE LEIGHTON BUILDING LE7 UNIVERSITY OF CENTRAL LANCASHIRE PRESTON, PR1 2HE, UKE-mail:sanscombe@uclan.ac.uk
FRANZ-VIKTOR KUHLMANN
Affiliation:
INSTITUTE OF MATHEMATICS UNIVERSITY OF SILESIA UL. BANKOWA 14, 40-007KATOWICE POLANDE-mail:fvk@math.us.edu.pl
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Abstract

We extend the characterization of extremal valued fields given in [2] to the missing case of valued fields of mixed characteristic with perfect residue field. This leads to a complete characterization of the tame valued fields that are extremal. The key to the proof is a model theoretic result about tame valued fields in mixed characteristic. Further, we prove that in an extremal valued field of finite p-degree, the images of all additive polynomials have the optimal approximation property. This fact can be used to improve the axiom system that is suggested in [8] for the elementary theory of Laurent series fields over finite fields. Finally we give examples that demonstrate the problems we are facing when we try to characterize the extremal valued fields with imperfect residue fields. To this end, we describe several ways of constructing extremal valued fields; in particular, we show that in every ℵ1 saturated valued field the valuation is a composition of extremal valuations of rank 1.

Keywords

valuationpolynomialmodel theory of valued fieldsextremal valued fieldtame valued field
Type
Articles
Information
The Journal of Symbolic Logic , Volume 81 , Issue 2 , June 2016 , pp. 400 - 416
Copyright
Copyright © The Association for Symbolic Logic 2016 

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References

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