Let $k$ be a field of character $p>0$ and consider $k(t) \subset k((t))$.
(1) Show that the field extension $k((t)) / k(t)$ is transcendent al, i.e., $k((t))$ contains at least one transcendental element ov er $k(t)$. (Hint: don't spend too much time on this question.)
(2) Let $\alpha \in k[[t]]$ which is transcendental over $k(t)$, and write $\beta=\alpha^p \in k[[t]]$. Let $K=k(t, \beta)$ and $L=k(t, \alpha)$ : both are subfields of $k((t))$. Let $A=k[[t]] \cap k(t, \beta)$. Determine the i ntegral closure $B$ of $A$ in $L$, and show that $A$ and $B$ are two discrete valuation rings.
(3) Is $B$ finitely generated over $A$ as a module? Justify your as sertion.