Regular closure of $\mathbb{Q}(t)$ Let $K$ be the algebraic closure of $\mathbb{Q}(t)$, so $\overline{\mathbb{Q}(t)} = K$. How to describe the subfield L of K which is the regular closure of $\mathbb{Q}(t)$, so $L\cap{\overline{\mathbb{Q}}}=\mathbb{Q}$ and $L$ is maximal with respect to this property and being algebraic over $\mathbb{Q}(t)$?

My guess is that $L$ might be the intersection of $K$ with the field of Puiseux series over $\mathbb Q$, i.e
$$ L \subset \mathbb Q\{\{t\} \} = \bigcup_{n\geq 1} \mathbb Q((t^{1/n})),$$
where $\mathbb Q((t^{1/n}))$ is the field of Laurent series in $t^{1/n}$ (power series allowing finitely many negative powers).
If you started with an algebraically closed ground field such as $\mathbb C$, then the algebraic closure of $\mathbb C((t))$ is the Puiseux series field $\mathbb C\{\{t\}\}$.
I couldn't find a nice description of the algebraic closure of $\mathbb C(t)$ though, and there isn't any answer given on MathOverflow either
