This thesis is a study of undecidability in some field theories. Specifically, we are interested
in geometrically oriented problems and have focused our attention in two directions along …

We study the existential theory of equicharacteristic henselian valued fields with a
distinguished uniformizer. In particular, assuming a weak consequence of resolution of …

We study various universal-existential fragments of first-order theories of fields, in particular
of function fields and of equicharacteristic henselian valued fields. For example we discuss …

We set up general machinery to study interpretations of fragments of theories. We then apply
this to existential fragments of theories of fields, and especially of henselian valued fields. As …

We construct an existentially undecidable complete discretely valued field of mixed
characteristic with existentially decidable residue field and decidable algebraic part …

We study the model theory of finitely ramified henselian valued fields of fixed initial
ramification, obtaining versions of the Ax–Kochen–Ershov principle as follows. We identify …

We extend the characterization of extremal valued fields given in [2] to the missing case of
valued fields of mixed characteristic with perfect residue field. This leads to a complete …

The thesis addresses certain problems in the model theory of henselian fields, with a special
focus on decidability. Some new methods are introduced along the way, which are of …

We prove a relative decidability result for perfectoid fields. This applies to show that the
fields ℚ p (p 1∕ p∞) and ℚ p (ζ p∞) are (existentially) decidable relative to the perfect hull …

In 1987, Pheidas showed that the field of Laurent series $\mathbb {F} _q ((t)) $ with a
constant for the indeterminate $ t $ and a predicate for the natural powers $\{t^ n\mid n> 0\} …