Symbolic Integration in Prime Characteristic

TL;DR

This paper investigates elementary extensions of differential fields in prime characteristic, revealing that unlike in characteristic zero, all elements have antiderivatives in some logarithmic extension.

Contribution

It demonstrates that in prime characteristic, every element in an elementary extension admits an antiderivative within a logarithmic extension, contrasting with classical results in characteristic zero.

Findings

All elements in elementary extensions have antiderivatives in logarithmic extensions.

Contrasts Liouville's classical result in characteristic zero.

Provides new insights into differential field extensions in prime characteristic.

Abstract

In this paper we study elementary extensions of differential fields in prime characteristic. In particular, we show that, in contrast to Liouville's result in characteristic zero, all elements of an elementary extension admit an antiderivative in some logarithmic extension.