# Reduced group schemes as iterative differential Galois groups

## TL;DR

This paper solves the inverse Galois problem for reduced algebraic group schemes over certain iterative differential fields, establishing conditions under which the problem has an affirmative answer in positive characteristic.

## Contribution

It demonstrates the solvability of the inverse Galois problem for reduced group schemes in positive characteristic and introduces the concept of equivalence of iterative derivations.

## Key findings

- Inverse Galois problem has an affirmative answer for reduced algebraic group schemes.
- Introduces the notion of equivalence of iterative derivations.
- Shows equivalence of inverse Galois problems under equivalent derivations.

## Abstract

This article is on the inverse Galois problem in Galois theory of linear iterative differential equations in positive characteristic. We show that it has an affirmative answer for reduced algebraic group schemes over any iterative differential field which is finitely generated over its algebraically closed field of constants. We also introduce the notion of equivalence of iterative derivations on a given field - a condition which implies that the inverse Galois problem over equivalent iterative derivations are equivalent.