Two examples concerning existential undecidability in fields

TL;DR

This paper constructs specific fields with contrasting decidability properties, demonstrating new examples of existential undecidability and decidability in valued fields, addressing open questions in the field.

Contribution

It provides the first examples of fields with mixed characteristic and residue fields with contrasting decidability properties, advancing understanding of existential undecidability in valued fields.

Findings

Constructed an existentially undecidable valued field with decidable residue field.

Built an existentially decidable field of positive characteristic with an undecidable finite extension.

Answered a question by Anscombe-Fehm regarding decidability in valued fields.

Abstract

We construct an existentially undecidable complete discretely valued field of mixed characteristic with existentially decidable residue field and decidable algebraic part, answering a question by Anscombe-Fehm in a strong way. Along the way, we construct an existentially decidable field of positive characteristic with an existentially undecidable finite extension, modifying a construction due to Kesavan Thanagopal.