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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.

Gevorg Hmayakyan
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    In general Galois theory provides the criterion that the Galois group of the polynomial is solvable. Computing the Galois group is not exactly trivial, but there are some methods that work for low degree polynomials, and also computing factorizations modulo a lot of small primes. – Tob Ernack Jan 04 '18 at 08:12
  • Thanks for answer. I am just looking a general method based on coefficients. – Gevorg Hmayakyan Jan 04 '18 at 08:13
  • What do you mean by based on coefficients? – Tob Ernack Jan 04 '18 at 08:24
  • Sorry I just have edited the question. The idea is to determine if solvable in radicals without computing the Galois group, but just from the coefficients directly. – Gevorg Hmayakyan Jan 04 '18 at 08:27
  • I mean there are some methods that work if you have some additional knowledge about the polynomial, such as particular root configurations. But I don't think there is a simple "formula" in terms of the coefficients that will tell you if a given polynomial is solvable or not in radicals depending on its value. The discriminant can tell you whether the Galois group is contained in $A_n$, but even the discriminant is a complicated function of the coefficients. – Tob Ernack Jan 04 '18 at 08:37
  • Could you please provide some more details about this method or may be some links? – Gevorg Hmayakyan Jan 04 '18 at 08:39
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    One method which I think is general is to find factorizations of the polynomial modulo many primes. This provides information on the cycle structures of elements of the Galois group, and after enough trials you will be able to determine the group. K. Conrad has some notes about Galois groups here: http://www.math.uconn.edu/~kconrad/blurbs/galoistheory/galoisSnAn.pdf – Tob Ernack Jan 04 '18 at 08:42
  • But it still makes use of the concept of Galois group. – Tob Ernack Jan 04 '18 at 08:42
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    Also see this post for proving that $x^5 - 4x +2$ is unsolvable by radicals – Tob Ernack Jan 04 '18 at 08:46

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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!

hardmath
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Jsevillamol
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