Valued fields with finitely many defect extensions of prime degree

TL;DR

This paper investigates valued fields with finitely many Artin-Schreier extensions, showing they are densely embedded in their perfect hull and deeply ramified, with implications for fields of both positive and mixed characteristic.

Contribution

It establishes new properties of valued fields with finitely many defect extensions, including density in the perfect hull and a partial analogue for mixed characteristic fields, filling gaps in previous classifications.

Findings

Valued fields with finitely many Artin-Schreier extensions are dense in their perfect hull.

Such fields are deeply ramified with p-divisible value groups and perfect residue fields.

Partial results are obtained for valued fields of mixed characteristic.

Abstract

We prove that a valued field of positive characteristic $p$ that has only finitely many distinct Artin-Schreier extensions (which is a property of infinite NTP$_2$ fields) is dense in its perfect hull. As a consequence, it is a deeply ramified field and has $p$-divisible value group and perfect residue field. Further, we prove a partial analogue for valued fields of mixed characteristic and observe an open problem about 1-units in this setting. Finally, we fill a gap that occurred in a proof in an earlier paper in which we first introduced a classification of Artin-Schreier defect extensions.