TITLE: Is a polynomial equation of degree $\ge 5$ not solvable by any way?
QUESTION [12 upvotes]: By Galois Theory an arbitrary polynomial equation of degree $\ge 5$ is not solvable  using radicals, unlike the polynomial equation of second degree which is solvable by radicals (because of the alternating  group of order 5, the symmetry group is not soluble).
Is this the only way an arbitrary polynomial equation can be (exactly) solved?
Does Galois Theory provide universal concept of general solvability?
Are there other methods except using radicals (for exact, not approximate or numerical solution)?
PS. A related question

REPLY [4 votes]: There are solvable quintics of course, but there is also a general solution of the quintic, using more than radicals, namely, using hypergemetric functions or theta functions.
See for example Quintic equation on MathWorld, and Quintic function, Bring radical on Wikipedia.
For the sextic, a Google search will show you several results, I'll put more references here if needed.