Let $K=\mathbb F_q((t))$. I want to prove that $K^{sep}$ is composite of $K^{sep}(p)$ and $K^{sep}(not \ p)$, where $K^{sep}(p)$ is maximal Galois extension of $K$ of exponent $p$, $K^{sep}(not \ p)$ is a maximal Galois extension of $K$ such that order of every element of $Gal(K^{sep}(not \ p)/K)$ is prime to $p$. Let $L/K$ be finite Galois extension, if it is totally ramified then $G=G_0$ and $G_0 = G_1 \rtimes G_0/G_1,G_1$ is $p$group, $\gcd(G_0/G_1,p)=1$ and we are done. But I can't understand what should I do if extension isn't totally ramified.

$\begingroup$ Three comments: first, $K^{sep}=L^{sep}$ where $L:=\overline{\mathbf{F}_q}((t))$. Second, every finite Galois extension of $L$ is totally ramified, with Galois group being a semidirect product of a normal $p$subgroup by a cyclic primeto$p$ subgroup. Third, such semidirect products do not correspond to composita of field in the way you suggest, since your condition forces the group to be a direct product rather than a semidirect product. $\endgroup$ – Michael Zieve May 21 '14 at 21:56

4$\begingroup$ @MichaelZieve: The field you call $L$ isn't algebraic over $K$, due to the intervention of formal power series. I think you mean $L$ to be the maximal unramified extension of $K$, which is to say the direct limit of the finite extensions $\mathbf{F}_{q^n}(\!(t)\!) = \mathbf{F}_{q^n} \otimes_{\mathbf{F}_q} K$ of $K$. $\endgroup$ – user76758 May 22 '14 at 4:08

1$\begingroup$ In other words, $\overline{\mathbf{F}_q}\otimes_{\mathbf{F}_q}\mathbf{F}_{q}(\!(t)\!)$. $\endgroup$ – Kevin Ventullo May 22 '14 at 7:09

$\begingroup$ @user76758: thanks for the correction. To salvage something of what I said, I just wanted to make the simple observation that the absolute Galois group of $\overline{\mathbf{F}_q}((t))$ is a subgroup of the absolute Galois group of $\mathbf{F}_q((t))$, and that we can say things about the former absolute Galois group. $\endgroup$ – Michael Zieve May 22 '14 at 8:38
This is not correct. If it was the compositum then you would get that the absolute Galois group is a product of pro$p$ group and pro$p$' groups (i.e. inverse limit of prime to $p$ finite groups). This is obviously wrong, since for example, if you take prime $\ell$ such that $p=1\pmod \ell$. Then the nonabelian group $C_p \rtimes C_\ell$ (with respect to an embedding of $C_{\ell}\to Aut(C_{p})$) is realizable over $\overline{\mathbb{F}}_q((t))$ but is not a product of $p$group and primeto$p$ group.
BTW: $C_m$ denotes a cyclic group of order $m$

$\begingroup$ Can you give me an example for an extension with Galois group equal to $C_\ell \rtimes C_p$, please? $\endgroup$ – user51074 May 21 '14 at 18:13

$\begingroup$ @MichaelZieve Thanks, I've corrected it $\endgroup$ – Lior BarySoroker May 22 '14 at 5:42

$\begingroup$ @user51074, for example take $K((t^{1/2}))/K((t))$ which is a $C_2$ extension and extend it further by adding a root of the ArtinSchreier equation $X^pX = t^{1/2}$. (Here $\ell=2$ and $p$ is an odd prime.) $\endgroup$ – Lior BarySoroker May 22 '14 at 5:48