Iterated Antiderivative 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 $F$.