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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.

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[1] Anscombe, S. and Fehm, A., The existential theory of equicharacteristic henselian valued fields,, 2015.
[2] Azgin, S., Kuhlmann, F.-V., and Pop, F., Characterization of extremal valued fields . Proceedings of the American Mathematical Society, vol. 140 (2012), pp. 15351547.
[3] van den Dries, L. and Kuhlmann, F.-V., Images of additive polynomials in q ((t)) have the optimal approximation property . Canadian Mathematical Bulletin, vol. 45 (2002), pp. 7179.
[4] Denef, J. and Schoutens, H., On the decidability of the existential theory of Fp [[t]], Valuation Theory and Its Applications, Vol. II (Saskatoon, SK, 1999), pp. 43–60, Fields Institute Communications, vol. 33, American Mathematical Society, Providence, RI, 2003.
[5] Durhan, S., Additive Polynomials over Perfect Fields , Valuation Theory in Interaction, Proceedings of the Second International Valuation Theory Conference, Segovia / ElEscorial, 2011, EMS Series of Congress Reports 2014.
[6] Kaplansky, I., Maximal fields with valuations I . Duke Mathematical Journal, vol. 9 (1942), pp. 303321.
[7] Kuhlmann, F.-V., Quantifier elimination for henselian fields relative to additive and multiplicative congruences . Israel Journal of Mathematics, vol. 85 (1994), pp. 277306.
[8] Kuhlmann, F.-V., Elementary properties of power series fields over finite fields , this Journal, vol. 66 (2001), pp. 771791.
[9] Kuhlmann, F.-V., Value groups, residue fields and bad places of rational function fields . Transactions of the American Mathematical Society, vol. 356 (2004), pp. 45594600.
[10] Kuhlmann, F.-V., On places of algebraic function fields in arbitrary characteristic . Advances in Mathematics, vol. 188 (2004), pp. 399424.
[11] Kuhlmann, F.-V., Dense subfields of Henselian fields, and integer parts , Logic in Tehran, Lecture Notes in Logic, vol. 26, pp. 204226, Association of Symbolic Logic, La Jolla, CA, 2006.
[12] Kuhlmann, F.-V., The algebra and model theory of tame valued fields . Journal für die Reine und Angewandte Mathematik, to appear. Preliminary version published in: Séminaire de Structures Algébriques Ordonnées, 81, Prépublications Paris 7 (2009).
[13] Pop, F., Embedding problems over large fields . Annals of Mathematics, vol. 144 (1996), pp. 134.
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The Journal of Symbolic Logic
  • ISSN: 0022-4812
  • EISSN: 1943-5886
  • URL: /core/journals/journal-of-symbolic-logic
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