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?

Answer to edited question: Yes and it works for arbitrary field $k((T))$ if either $k$ is of characteristic 0 or when the ramification index is not divisible by characteristic of $k$, assuming that your notation $k((T))$ refers to power series ring with coefficients in $k$. This is theorem 6 of chapter 4 section 1 (page 264) in Borevich-Shafarevich's Number Theory.
