Eisenstein theorem for algebraic power series in positive characteristic

Question

Eisenstein proved that if a power series $\sum_{n\ge0}a_nz^n$ over $\mathbb C$ is algebraic over $\overline{\mathbb Q}(z)$, then it exists positive integers $a$ and $b$ such that for all $n\in\mathbb N$, $ba^na_n$ is an algebraic integer. Obviously, adapting the proof shows that the following is also true in positive characteristic:

Let $\Omega$ be a completion of an algebraic closure of $\mathbb F_q\left(\left(\frac1T\right)\right)$ for the topology induced by the valuation $-\deg$. Let $f(z)=\sum_{n\ge0}a_nz^n$ be a power series over $\Omega$. If $f$ is algebraic over $\overline{\mathbb F_q(T)}(z)$, then it exists $P,Q\in\mathbb F_q[T]$ such that for every $n\in\mathbb N$ $QP^na_n$ belongs to the integral closure of $\mathbb F_q[T]$ in $\overline{\mathbb F_q(T)}$.

Does anyone know where I could find this theorem explicitly enounced and better proved.

Thanks in advance

Answer 1

After two years of research, I found a reference: Lemma 2.1 of

TRANSCENDENCE OF GENERATING FUNCTIONS WHOSE COEFFICIENTS ARE MULTIPLICATIVE

JASON P. BELL, NILS BRUIN and MICHAEL COONS Transactions of the American Mathematical Society Vol. 364, No. 2 (FEBRUARY 2012), pp. 933-959