Let $f(z)=\sum_{n\geq 0}a_n z^n$ be a Taylor series with rational coefficients with infinitely non-zero $a_n$ which converges
in a small neighboorhood around $0$. Furthermore, assume that
\begin{align*}
f(z)=\frac{P(z)}{Q(z)},
\end{align*}
where $P(z)$ and $Q(z)$ are *coprime monic complex polynomials*. By developing $\frac{P(z)}{Q(z)}$ as a power sereis around $0$ and comparing it with $f(z)$ we obtain infinitely many polynomial equations in the roots of $P(z)$ and $Q(z)$ which are equal to rational numbers so this seems to force the roots of $P(z)$ and $Q(z)$ to be algebraic numbers.

Q: How does one prove this rigourously?