A Hahn-like construction for finitely ramified valued fields

Explicit constructions of models of the theory of a valued field are useful tools for understanding its model theory. Since Kaplansky’s work, it has been a topic of interest to characterize value fields in terms of fields of power series. In particular, Kaplansky proved that, under certain assumptions, an equicharacteristic valued field is isomorphic to a Hahn field. In this talk, we show that in the mixed characteristic case, assuming the Continuum Hypothesis, we can provide a characterization, in terms of power series, of pseudo-complete finitely ramified valued fields with a fixed residue field k and valued in a Z-group G, using a Hahn-like construction with coefficients in a finite extension of the Cohen field C(k) of k. In this construction, the elements of the field are “twisted” power series, i.e. powers series whose product is defined by having an extra factor, given by the cross-section and a 2-cocycle determined via the value group. This generalizes a result by Ax and Kochen, who characterize pseudo-complete valued fields elementarily equivalent to the field of p-adic numbers Q_p. If time permits, we will see some consequences of this characterization regarding the problem of lifting automorphisms of the residue field and the value group to automorphisms of the valued field in the mixed characteristic case.