I am currently reading the book "Differential Galois Theory" by Springer and Van der Put. In the process of establishing that the field fixed by the differential galois group of a picard vessiot extension is the base field, the authors remark that:
If $G$ is a differential galois group for a Picard-Vessiot extension of fields associated to a differential equation over the base field and $H$ be a subgroup of the differential Galois group, then $L^{H}= L^{\overline{H}}$, where $\overline{H}$ is the Zariski closure.
Can anyone help me to see why this is very obvious, as that is what the authors have remarked. Thanks.