Infinitesimal group schemes as iterative differential Galois groups

TL;DR

This paper characterizes when infinitesimal group schemes can serve as Galois groups in the context of iterative differential fields in positive characteristic, solving the inverse Galois problem for these schemes.

Contribution

It provides a necessary and sufficient condition for infinitesimal group schemes to be Galois groups of Picard-Vessiot extensions over ID-fields.

Findings

Provides a criterion for realizing infinitesimal group schemes as Galois groups.

Solves the inverse ID-Galois problem for infinitesimal group schemes.

Advances understanding of Galois theory in positive characteristic.

Abstract

This article is concerned with Galois theory for iterative differential fields (ID-fields) in positive characteristic. More precisely, we consider purely inseparable Picard-Vessiot extensions, because these are the ones having an infinitesimal group scheme as iterative differential Galois group. In this article we prove a necessary and sufficient condition to decide whether an infinitesimal group scheme occurs as Galois group scheme of a Picard-Vessiot extension over a given ID-field or not. In particular, this solves the inverse ID-Galois problem for infinitesimal group schemes.