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Let $q$ be a prime power, $\mathbb{F}_q$ the field with $q$ elements and $f \in \mathbb{C}_\infty$ be of the form $f = \prod_{i=1}^\infty f_i$ with $f_i \in \mathbb{F}_q(X)$ (here $\mathbb{C}_\infty$ is the completion of an algebraic closure of $\mathbb{F}_q((X^{-1}))$). Denote with $f'$ the formal derivative of $f$. Is it true, that $f'$ is algebraic (over $\mathbb{F}_q(X)$) if $f$ is algebraic?

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closed as off-topic by Adam Hughes, John B, colormegone, Edward Jiang, Daniel W. Farlow Apr 29 '16 at 0:56

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  • $\begingroup$ How do you define that "infinite product" of polynomials? $\endgroup$ – Alex M. Apr 29 '16 at 19:36
  • $\begingroup$ @Alex I edited the question to clarify the assumptions $\endgroup$ – Martin Apr 30 '16 at 6:45
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Consider the factorization of $f$: $$f(X) = \Pi_{i=1}^n (X-\alpha_i).$$ By the Leibniz Rule (which even formal derivatives satisfy) it follows that $f'$ is algebraic if and only if $f$ is.

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