Consider $\theta:\mathbb F_q(T)\mapsto\mathbb F_q(T)$ defined by $\theta(Q)=Q(T^q)$. It is a morphism of fields. Let $P$ be an irreducible polynomial of $\mathbb F_q[T]$. Then, $\theta$ can be uniquely extended to $\mathbb F_q(T)_P$ by continuity (denote it $\theta$ again), where $\mathbb F_q(T)_P$ is the completion of $\mathbb F_q(T)$ for the topology induced by the $P$-valuation . It is a morphism of fields yet and for every $x\in K_P$, one has $\theta(x)=x^q$. My question: can $\theta$ be extended continuously to $\Omega_P$, the topological closure of an algebraic closure $\overline{K_P}$ of $K_p$ such that $\theta$ is a morphism of fields in $\overline{K_P}$ and $\theta(x)=x^q$ for any element $x\in\overline{K_P}$?
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