A good deal of the arithmetic of a field can be expressed by sentences in the first-order language of rings. The theories
of the characteristic zero local fields have been axiomatized and are decidable: in the case of $Q_p$ and its finite extensions,
Ax, Kochen, and (independently) Ershov, gave complete axiomatizations that are centred on a formalization of Hensel’s
Lemma. In fact the theory of any field of characteristic zero which is complete with respect to a non-archimedean
valuation can be likewise axiomatized.

I will explain recent joint work with Jahnke, and also with Dittmann and Jahnke, in which we extend the classical
work on these theories to include the case of imperfect residue fields. In particular we show that “Hilbert’s Tenth
Problem” (H10) in these fields (i.e. the problem of effectively determining whether a given Diophantine equation has
solutions) is solvable if and only if the analogous problem is solvable on a structure we define on the residue field. This
follows a pattern of such “transfer” results for H10 — established for valued fields of positive characteristic in earlier
work with Fehm — although in the current case we really need the extra structure.

I will describe these results, focusing on the extent to which they depend (or not) on the residue field. If there is
time I will discuss the aforementioned H10 transfer for complete valued fields in positive characteristic, including more
recent uniform aspects.

I will not assume a background in logic.