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Let $P$ be an irreducible polynomial of $\mathbb F_q[T]$, $(u_n)_n$ be an infinite sequence of distinct elements of $\mathbb N_0$. Does there exist infinitely many multiples of $P$ in $\mathrm{Vect}_{\mathbb F_q}(T^{u_n}\mid n\in\mathbb N_0)$?

Thanks in advance.

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  • $\begingroup$ to be clear, you mean the vector space spanned by the $T^{u_n}$? $\endgroup$
    – Kevin Casto
    Commented Apr 13, 2022 at 19:35
  • $\begingroup$ yes, I meant that. $\endgroup$
    – joaopa
    Commented Apr 13, 2022 at 19:40
  • $\begingroup$ What's the motivation for this question? $\endgroup$
    – Will Sawin
    Commented Apr 13, 2022 at 19:40
  • $\begingroup$ I try to build a sequence of elements with small $P$-adic norm in this vector space. $\endgroup$
    – joaopa
    Commented Apr 13, 2022 at 19:43

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Sure. Pick a root $\gamma$ of $P$ lying in some field extension $\mathbb{F_{q'}}$. We have a $\mathbb{F}_q$-linear map from this space of polynomials to $\mathbb{F_{q'}}$ given by evaluating a polynomial at $\gamma$. This is going from an infinite dimensional space to a finite one, so it has a big kernel.

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