On algebraically maximal valued fields that are not defectless

TL;DR

This paper constructs algebraically maximal valued fields in various characteristics and ranks that admit specific inseparable and separable extensions with defect, expanding understanding of defect phenomena in valuation theory.

Contribution

It introduces new examples of algebraically maximal valued fields with controlled defect behavior in different characteristics and ranks.

Findings

Existence of algebraically maximal fields with defect in characteristic p and 0

Construction of rank 2 algebraically maximal fields with defect

Fields admit separable or inseparable extensions of degree p^2 with defect

Abstract

We use a known example of an algebraically maximal discretely valued field of positive characteristic $p$ which admits purely inseparable extensions of degree $p^2$ with defect $p$ to construct algebraically maximal valued fields of characteristic $p$ as well as of characteristic $0$ and of rank 2 which admit separable extensions of degree $p^2$ with defect $p$.