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I investigated a family of polynomials given by for $n>1$:

$x^n+(x+1)^n+...+(x+n-1)^n=(x+n)^n$

And I found, with the help of Magma, that the Galois group is $S_n$ for $n$ up to a hundred. The small examples up to $n=10$ also have a unique positive solution and I think proving this fact will not be hard.
But my main question is, whether the Galois group is $S_n$ for all $n$. I have almost no idea how to compute the Galois group, but maybe the special form of the equation will allow a moderately hard solution.
I am also interested in a softer problem: Is it true, that for $n>5$ the Galois group of the equation above is non-solvable?

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  • $\begingroup$ Can you prove that the polynomial is irreducible for all $n$? $\endgroup$
    – user6976
    Apr 18, 2018 at 0:48

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