TITLE: Galois group over $p$-adic numbers
QUESTION [25 upvotes]: Can one describe explicitly the Galois group  $G=\operatorname{Gal}(\overline{\mathbb Q_p}/\mathbb Q_p)$? 
I only know the most basic stuff: unramified extensions of $\mathbb Q_p$ are equivalent to extensions of $\mathbb F_p$, so the Galois group of the unramified part is $\hat{\mathbb Z}$, generated by the Frobenius automorphism. 
Then I have a tame knowledge of tamely ramified extensions, given by adjoining $p^{1/n}$, $(p,n)=1$. Elements of $G$ send $p^{1/n}$ to $\alpha p^{1/n}$, $\alpha^n=1$. Alltogether, I saw the tamely ramified Galois group described neatly as the profinite group with generators $\sigma,\tau$ ($\sigma$ a lift of Frobenius) and relation $\sigma\tau\sigma^{-1}=\tau^p$.
But I know wildly nothing about wildly ramified extensions, i.e. about the kernel of the projection $G\to\langle \sigma,\tau|\sigma\tau\sigma^{-1}=\tau^p\rangle$ (besides perhaps that it's a pro-$p$ group).

REPLY [7 votes]: Your question is about the group $P=\mathrm{Gal}(\bar{\mathbf Q}_p|V)$, where $V$ is the maximal tamely ramified extension of $\mathbf{Q}_p$. In his On Galois Groups of Local Fields, Transactions of the American Mathematical Society Vol. 80, No. 2 (Nov., 1955), pp. 448-469, Iwasawa proved that $P$ is the pro-$p$ completion of the free group on countably many free generators (Theorem 2(ii)).  He also proved that the short exact sequence
$$
1\to P\to \mathrm{Gal}(\bar{\mathbf Q}_p|\mathbf{Q}_p) \to \Gamma\to1
$$
splits (Theorem 2(iii)), where $\Gamma=\mathrm{Gal}(V|\mathbf{Q}_p)$. If we choose a splitting, we get the conjugation action of $\Gamma$ on $P$.  Iwasawa did not describe this action but he describes the resulting action on the maximal abelian quotient of $P$ using local class field theory ($\S$3.3). 
His results are valid not just for $\mathbf{Q}_p$ but for any finite extension of $\mathbf{Q}_p$.