The Cyclic Vector Lemma

Let $F$ be a differential field of characteristic zero with algebraically closed constant field $C$. Let $E$ be a Picard–Vessiot closure of $F$, $R\subset E$ its Picard–Vessiot ring and $\mathrm{\Pi }$ the differential Galois group of $E$ over $F$. Let $V$ be a differential $F$ module, finite dimensional as an $F$ vector space. Then $V$ is singly generated as a differential $F$ module if and only if there is a $\mathrm{\Pi }$ module injection ${\text{Hom}}_F^{\text{diff}}(V,R)\to R$. If $C\mathrm{\ne }F$ such an injection always exists.