Extensions of $\mathbb{Q}((T))$ and $\mathbb{F}_p((T))$ Okay, I'm having some trouble finding good references for this, so here goes:
Is every finite extension of $\mathbb{Q}((T))$ isomorphic to $K((T^{1/e}))$ where $K$ is finite over $\mathbb{Q}$, and $e$ is an integer?
Is every finite extension of $\mathbb{F}_p((T))$ isomorphic to $\mathbb{F}_q((T^{1/e}))$, $q = p^r$?
Examples, proofs, and references would be appreciated.
EDIT: Julian pointed out that obviously the answer is no. So my question is now: Is it true that for every finite extension $L$ of $\mathbb{Q}((T))$ or $\mathbb{F}_p((T))$, there exists an extension of $L$ which is of the form I described above?

For $k$ an arbitrary field and $u\in k$ not an $e$-th power, $k((\;(uT)^{1/e}\;))$ is an extension of $k((T))$ that is not of the form you describe.
