Criteria to determine if equation is solvable in radicals? Assume we have an equation:
$$\sum_{i=0}^p a_ix^i=0$$
where p is prime and the left side polynomial is irreducible. Is there a method or algorithm to determine if this equation is solvable in radicals?

EDITED

I mean is there a general method to determine if it is solvable in radicals without computing an appropriate Galois group.
There are in fact some criteria that allow you to determine if the polynomial is solvable by radicals (from here thereon solvable) in some special cases!

The plainest one is that if the degree is less than 5, the polynomial is solvable.

If your polynomial is of the form $f(T)=g(T^k)$ and $g$ is solvable, then $f$ is solvable.

A more exciting one: if $f$ is irreducible of prime odd degree (as it is the case), then $f$ is solvable iff its splitting field is generated by any two of its roots.

In particular, this means that if $f$ has at least two real roots and one complex root then it is NOT solvable.

If you want a truly general algorithm, I am afraid I have no better way than jumping through the hoop of Galois theory and determining if its Galois group is solvable.



EDIT: I did some digging, and S. Landau and G. Miller have you covered:


  A polynomial time algorithm is presented for the founding question of Galois theory: determining solvability by radicals of a monic irreducible polynomial over the integers. 


Their approach passes through Galois theory though, but hey polynomial time!
I'm not sure why you are considering only prime $p $...

It is proved in Galois theory that polynomials of degree  $5$ and up are not solvable by radicals...  that is there is no closed form that works for all such polynomials...