A criterion for $p$-closedness of derivations in dimension two

TL;DR

This paper introduces a practical criterion for identifying certain derivations in two-dimensional algebraic structures, enhancing the understanding of purely inseparable field extensions in positive characteristic.

Contribution

It provides a new, efficient criterion for p-closed derivations in dimension two, advancing the algebraic theory of inseparable extensions.

Findings

The criterion simplifies the identification of p-closed derivations.

Examples demonstrate the criterion's effectiveness.

The approach advances understanding of inseparable field extensions.

Abstract

Jacobson developed a counterpart of Galois theory for purely inseparable field extensions in positive characteristic. In his theory, a certain type of derivations replace the role of the generators of Galois groups. This article provides a convenient criterion for determining such derivations in dimension two. We also present examples demonstrating the efficiency of our criterion.