# 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.

## Key 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.