Galois theory tells us that that some polynomials \vskip .1in $f(x) = x^n + a_1 x^{n-1} + ... + a_{n-1} x + a_n$ \vskip .1in \noindent of degree $n > 4$ cannot be solved in radicals. Equivalently, some $S_n$-covers cannot be split by a solvable base extension. J. Tits asked whether an analogous assertion remains true if $S_n$ is replaced by a connected group $G$. In this talk I will discuss the background of this problem and recent results (obtained jointly with V. Chernousov and $P$. Gille) which indicate that solvability in radical may, indeed be possible in this setting. In particular, I will explain a connection we found between Tits' question and a variant Hilbert's 13th problem.