Iterated Antiderivative Extensions

TL;DR

This paper proves that subfields of iterated antiderivative extensions over a differential field with algebraically closed constants are themselves iterated antiderivative extensions, clarifying their structural properties.

Contribution

It establishes that any differential subfield containing the base field within an iterated antiderivative extension retains the same extension type.

Findings

Subfields of iterated antiderivative extensions are also iterated antiderivative extensions.

The structure of such extensions is preserved under taking differential subfields.

Provides a foundational result for the theory of Liouvillian extensions.

Abstract

Let $F$ be a characteristic zero differential field with an algebraically closed field of constants and let $E$ be a no new constants extension of $F$. We say that $E$ is an \textsl{iterated antiderivative extension} of $F$ if $E$ is a liouvillian extension of $F$ obtained by adjoining antiderivatives alone. In this article, we will show that if $E$ is an iterated antiderivative extension of $F$ and $K$ is a differential subfield of $E$ that contains $F$ then $K$ is an iterated antiderivative extension of $F$.