Prof. Anna De Mase, from Università degli Studi della Campania “Luigi Vanviltelli”, will give a seminar on Monday 8 May 2023 at 14:30 in Sala Riunioni, DipMat.
Title: A characterization for saturated models of a finite extension of Qp: the “twisted” power series.
Abstract: In 1965, Ax and Kochen gave a complete axiomatization of the theory of the class of formally p-adic fields, by having the following properties:
– the value group is a Z-group;
– the residue field is Fp;
– v(p) is the minimal positive element of the value group.
Thus, a valued field K is formally p-adic if and only if it is elementarily equivalent to Qp. Moreover, assuming the Continuum Hypothesis, they give a characterization of ω-pseudo complete formally p-adic fields with a fixed value group G of cardinality ℵ1, using a Hahn-like construction over Qp that preserves the residue field. In this construction, the elements of the field are “twisted” power series over Qp, i.e. power series whose product is defined by having a power of p as an extra factor, and whose exponent is given by a 2-cocycle defined through the value group. In this talk, we generalize this construction to the more general setting of mixed characteristic henselian valued fields with finite ramification and valued in a discrete ordered abelian group G. Then we show that an analogous characterization holds for ω-pseudo complete valued fields elementarily equivalent to some extensions of the p-adics.
