# Correction and notes to the paper "A classification of Artin-Schreier defect extensions and characterizations of defectless fields"

## TL;DR

This paper corrects a lemma in a previous work on Artin-Schreier defect extensions, clarifies its impact, and discusses recent generalizations of the classification of defect extensions.

## Contribution

It rectifies a specific mistake in earlier classification work and introduces new results on linearly disjoint field extensions relevant to defect theory.

## Key findings

- The correction does not affect the original results.
- New results on linearly disjoint extensions are presented.
- An example shows the importance of separability assumptions.

## Abstract

We correct a mistake in a lemma in the paper cited in the title and show that it did not affect any of the other results of the paper. To this end we prove results on linearly disjoint field extensions that do not seem to be commonly known. We give an example to show that a separability assumption in one of these results cannot be dropped (doing so had led to the mistake). Further, we discuss recent generalizations of the original classification of defect extensions.