2 problems

Let $\mathbf{A}=\mathbb{F}_q[t]$, $A=\mathbb{F}_q[\theta]$, and $K=\operatorname{Frac}(A)=\mathbb{F}_q(\theta)$. Let $E$ be a finite extension of $K$ and let…

Let $K$ be a finite separable extension of $\mathbf{F}_q(t)$, let $A=\mathbf{F}_q[t]$, and let $K_\infty$ and $O_K$ denote the completions and the ring of integers used in the defi…