Let $f(x) = a_{2n} x^{2n} + a_{2n-1} x^{2n-1} + \cdots + a_0$ be a polynomial with integer coefficients and irreducible over $\mathbb{Q}$. For $n \geq 3$, $f$ is generically unsolvable by radicals. Indeed, most irreducible polynomials of degree $2n$ have Galois group $S_{2n}$, which is unsolvable for $n \geq 3$. I am looking for a characterization, possibly in terms of the roots of $f$, for when $f$ is solvable by radicals.

For the $n = 3$ case, a very satisfactory answer is given in this paper: http://www.sciencedirect.com/science/article/pii/S002186930098428X

In particular, an answer such as "the Galois group of $f$ is solvable if and only if one of a small list of Galois resolvents has a rational root" would be very satisfactory.