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valued fields with an automorphism

Journal of the Institute of Mathematics of Jussieu (1)

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SOME PROPERTIES OF ANALYTIC DIFFERENCE VALUED FIELDS
Part of
Silvain Rideau
Journal:

Journal of the Institute of Mathematics of Jussieu / Volume 16 / Issue 3 / June 2017

Published online by Cambridge University Press:

29 May 2015, pp. 447-499

Print publication:

June 2017
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We prove field quantifier elimination for valued fields endowed with both an analytic structure that is $\unicode[STIX]{x1D70E}$-Henselian and an automorphism that is $\unicode[STIX]{x1D70E}$-Henselian. From this result we can deduce various Ax–Kochen–Eršov type results with respect to completeness and the independence property. The main example we are interested in is the field of Witt vectors on the algebraic closure of $\mathbb{F}_{p}$ endowed with its natural analytic structure and the lifting of the Frobenius. It turns out we can give a (reasonable) axiomatization of its first-order theory and that this theory does not have the independence property.