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).
Best Answer
Even though every extension of $\mathbb{Q}_p$ is solvable, I don't think one can write down a formula for the solution to a general quintic in terms of radicals; if one could, then the $S_5$-extension $\mathbb{Q}_p(r_1,...,r_5)/\mathbb{Q}_p(e_1,...,e_5)$ would be solvable, where the $e_i$ are the elementary symmetric polynomials in the $r_i$.