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?
 A: 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.
A: For a finite field like $k=\Bbb F_p$, your conjecture is nowhere near true. Look at the extension $k((T^{1/e}))\supset k((T))$, and write $e=p^mg$, with $g$ prime to $p$. Then you can adjoin $T^{1/g}$ first and then the $p^m$-th root of that. The first extension is tamely ramified, the second is totally inseparable. But there are loads of wildly ramified separable extensions: these have degree $p^n$ but are separable. Here’s the simplest example:
Consider the ring $R=k[[T]]$, and the automorphism $\varphi$ of $R$ that takes $T$ to $T/(1-T)=T+T^2+T^3+\cdots$. Think of this as a fractional-linear transformation with matrix
$$
\begin{pmatrix}
1&0\\-1&1
\end{pmatrix}\,,
$$
and you see that the $p$-fold iterate of $\varphi$ is identity. Thus $\langle\varphi\rangle$ is a cyclic group of automorphisms of $k((T))$, with a fixed field that is necessarily of form $k((t))$, indeed you see that you can take $t=T^p/(1-T^{p-1})$. The extension $K((T))\supset k((t))$ is necessarily Galois with group $\langle\varphi\rangle$, wildly ramified, and certainly not expressible in the form you hoped, and even more certainly not covered, as you say by an extension you hoped for.
I should also say that the abelian extensions of $k((T))$ are described by Local Class Field Theory, and you can look that up.
A: 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.
