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Additive polynomials and their role in the model theory of valued fields

Published online by Cambridge University Press:  30 March 2017

By
Franz-Viktor Kuhlmann
Edited by
Ali Enayat ,
Iraj Kalantari  and
Mojtaba Moniri
Ali Enayat
Affiliation:
American University, Washington DC
Iraj Kalantari
Affiliation:
Western Illinois University
Mojtaba Moniri
Affiliation:
Tarbiat Modares University, Tehran, Iran
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Summary

Abstract.We discuss the role of additive polynomials and p-polynomials in the theory of valued fields of positive characteristic and in their model theory. We outline the basic properties of rings of additive polynomials and discuss properties of valued fields of positive characteristic as modules over such rings. We prove the existence of Frobenius-closed bases of algebraic function fields F |K in one variable and deduce that F/K is a free module over the ring of additive polynomials with coefficients in K. Finally, we prove that every minimal purely wild extension of a henselian valued field is generated by a p-polynomial.

Introduction.This paper is to some extent a continuation of my introductive and programmatic paper [Ku3]. In that paper I pointed out that the ramification theoretical defect of finite extensions of valued fields is responsible for the problems we have when we deal with the model theory of valued fields, or try to prove local uniformization in positive characteristic.

In the present paper I will discuss the connection between the defect and additive polynomials. I will state and prove basic facts about additive polynomials and then treat several instances where they enter the theory of valued fields in an essential way that is particularly interesting for model theorists and algebraic geometers. I will show that non-commutative structures (skew polynomial rings) play an essential role in the structure theory of valued fields in positive characteristic. Further, I will state the main open questions. I will also include some exercises.

In the next section, I will give an introduction to additive polynomials and describe the reasons for their importance in the model theory of valued fields. For the convenience of the reader, I outline the characterizations of additive polynomials in Section 3 and the basic properties of rings of additive polynomials in Section 4. For more information on additive polynomials, the reader may consult [Go], cf. also [O1], [O2], [Wh1], [Wh2], [Ku4], [Dr–Ku]. The remaining sections of this paper will then be devoted to the proofs of some of the main theorems stated in Section 2.

Type
Chapter
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Logic in Tehran , pp. 160 - 203
Publisher: Cambridge University Press
Print publication year: 2006

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