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?
Best Answer
Let there be two fields $k\subset K$, and let $f\in k[[x]]$ be a formal power series with coefficients in $k$. If $f\in K(x)$ (rational functions with coefficients in $K$) then $f\in k(x)$. A proof of this is given in J.S. Milne's notes on Etale Cohomology (lemma 27.9).