Q:

Galois Group of Polynomial

I would like to compute the Galois group of the Polynomial 
$f(x)=x^5-5x^4 +10 x^3 - 10 x^2 - 135 x + 131\in\mathbb{Q}[x] $
I already know that it is irreducible in $\mathbb{Q}[x]$ via Eisenstein's criterion, $ f(x-1)$ and $p=5$, but have no idea how to proceed.
Thank you very much!

A:

As the polynomial is irreducible of degree $5$, the group is a subgroup of $S_5$ containing a $5$-cycle. If it has exactly $2$ non-real roots, then the group has a transposition coming from complex conjugation, and you should be able to take it from there. If it has $4$ non-real roots, it will take some more work. 

