Elementarity of the inclusion F_p(t)^h into F_p((t))

Determine whether the natural inclusion of valued fields (ℱ_p(t)^h, v_t) → (ℱ_p((t)), v_t) is an elementary embedding in the language of valued fields, i.e., whether ℱ_p(t)^h is an elementary substructure of ℱ_p((t)).

Background

The paper develops Ax–Kochen/Ershov principles for deeply ramified fields of positive characteristic with fixed degree of imperfection. As an application, the authors consider a known problem concerning the model theory of local function fields: whether the henselization ℱ_p(t)^h embeds elementarily into the Laurent series field ℱ_p((t)).

While the paper does not resolve this problem, it proves elementarity results for a broad class of related extensions: if K is a deeply ramified algebraic extension of ℱ_p(t)^h, then K embeds elementarily into K·ℱ_p((t)). In particular, this yields an elementary embedding for the Artin–Schreier closures. The original question about ℱ_p(t)^h itself, however, remains open.