TITLE: Solubility of the quintic?
QUESTION [26 upvotes]: Over the p-adics, every Galois group is solvable.  Does this imply that the quintic (and higher-order polynomials for that matter) can be solved by radicals over $\mathbb{Q}_p$?
EDIT: The original place I learned that the p-adic galois groups were solvable was in Milne's Algebraic Number Theory text (Chapter 7, Cor 7.59).  
As was pointed out the comments, I should clarify that I meant to ask 2 questions.  Namely, whether the general quintic can be solved by radicals in this context (still no) and whether any given one can be (which I now believe is yes).

REPLY [4 votes]: Try Lazard (Daniel),
Solving quintics by radicals, in The legacy of Niels Henrik Abel, 207–225, Springer, Berlin, 2004.
MR2077574 (2005g:12002) says : 
Let $F$ be a field of characteristic different from 2 and 5. Let $f$ be a univariate irreducible polynomial of degree 5 over $F$. The polynomial $f$ is said to be solvable by radicals if the Galois group over $F$ of the field generated by all the roots of $f$ is solvable. In the paper the author gives a formula for solving by radicals any polynomial $f$ of degree 5 which is solvable by radicals. The field extension which is generated by the radicals which appear in the result is always minimal, when only one root is produced, as well as when all roots are given. This formula has been implemented in Maple.
Reviewed by Jerzy Urbanowicz.