Regular inverse Galois problem for Q(t)

Question

I am interest in the Regular Inverse Galois Problem for $\mathbb{Q}(t)$. Reading Serre "Topics in Galois theory" I understood how to realize finite group $G$ as Galois group over $\mathbb{C}(t)$ and $\bar{\mathbb{Q}}(t)$. I also basically understood how to descend from $\bar{\mathbb{Q}}(t)$ to $\mathbb{Q}(t)$. However my question is follows: if we descended to $\mathbb{Q}(t)$, why the Galois extension is automatically regular? Regular means $K \cap \bar{\mathbb{Q}} = \mathbb{Q}$ where $K/\mathbf{Q}(t)$ is a extension. I mean in the proof Serre constructed the extension but didn't say anything about the regularity. But it seem to be important since we call it "Regular Inverse Galois Problem"!