Let $\mathcal{M}$ be the field of meromorphic functions of one (complex) variable and $w = w(z)$ an analytic function satisfying a polynomial equation

$P(w; z) := w^n + a_{n-1}(z) w^{n-1} + \cdots + a_1(z) w + a_0(z) = 0$,

where $a_0(z), \ldots, a_{n-1}(z)$ are in $\mathcal{M}$ (actually, is suffices to consider the case where $a_j(z)$ is entire for $j = 0, \ldots, n-1$).

Suppose $w(z)$ has finitely many branch points.

In the (hyperelliptic) case $n=2$ it is clear that $\mathcal{M}(w) = \mathcal{M}(\sqrt{Q})$, where $Q(z)$ is a polynomial.

Is it always true that, under the above assumptions we have that $\mathcal{M}(w) = \mathcal{M}(\beta)$, where $\beta(z)$ is some algebraic function?