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Let $P$ be a an irreducible polynomial of $k:=\mathbb F_q(T)$, $\Omega_P$ be the completion of an algebraic closure $\overline{k_P}$ of $k_P$, the completion of $k$ for the topology induced by the $P$-adique valuation. Consider an algebraic closure $\overline k$ of $k$ in $\Omega_P$ and $\alpha\in\overline k$. Does it exist a continuous $\mathbb F_q$-morphism $\sigma$ of $\Omega_P$ such that $\sigma(T)=T+\xi$ with $\xi\in\mathbb F_q$ and $\sigma(\alpha)=\alpha$. If not in a whole generality, can one determine the $\alpha's$ such that the problem admits an answer?

Let $P$ be a an irreducible polynomial of $k:=\mathbb F_q(T)$, $\Omega_P$ be the completion of an algebraic closure $\overline{k_P}$ of $k_P$, the completion of $k$ for the topology induced by the $P$-adique valuation. Consider an algebraic closure $\overline k$ of $k$ in $\Omega_P$ and $\alpha\in\overline k$. Does it exist a continuous $\mathbb F_q$-morphism $\sigma$ of $\Omega_P$ such that $\sigma(T)=T+\xi$ with $\xi\in\mathbb F_q$ and $\sigma(\alpha)=\alpha$. If not in a whole generality, can one determine the $\alpha's$ such that the problem admits a positive answer?

Let $P$ be a an irreducible polynomial of $k:=\mathbb F_q(T)$, $\Omega$ be the completion of an algebraic closure $\overline{k_P}$ of $k_P$, the completion of $k$ for the topology induced by the $P$-adique valuation. Consider an algebraic closure $\overline k$ of $k$ in $\Omega_P$ and $\alpha\in\overline k$. Does it exist a continuous $\mathbb F_q$-morphism $\sigma$ of $\Omega_P$ such that $\sigma(T)=T+\xi$ with $\xi\in\mathbb F_q$ and $\sigma(\alpha)=\alpha$. If not in a whole generality, can one determine the $\alpha's$ such that the problem admits an answer?

Let $P$ be a an irreducible polynomial of $k:=\mathbb F_q(T)$, $\Omega_P$ be the completion of an algebraic closure $\overline{k_P}$ of $k_P$, the completion of $k$ for the topology induced by the $P$-adique valuation. Consider an algebraic closure $\overline k$ of $k$ in $\Omega_P$ and $\alpha\in\overline k$. Does it exist a continuous $\mathbb F_q$-morphism $\sigma$ of $\Omega_P$ such that $\sigma(T)=T+\xi$ with $\xi\in\mathbb F_q$ and $\sigma(\alpha)=\alpha$. If not in a whole generality, can one determine the $\alpha's$ such that the problem admits an answer?