Galois group of degree 7 polynomial

Question

From Bosch‘s Algebra (p. 368):

Determine the galois group of $$X^7 - 8X^5 - 4X^4 + 2X^3 - 4X^2 + 2 \in \mathbb Q[X]$$ and decide if it‘s solvable or not.

How does one find the galois group of that polynomial?

My usual way is to find the roots $\alpha_1, \ldots, \alpha_n$, find a splitting field $L = L(\alpha_1, \ldots, \alpha_n)$ and then study the structure of the galois group $\operatorname{Gal}(L/\mathbb Q) = \operatorname{Gal}(f)$ as a subgroup of $S_n$.

However that does not seem to work in this case since already finding roots seems difficult. What is a useful approach in this case?

Answer 1

Hint

1. Use Eisenstein's Criterion to show that $f$ is irreducible.
2. Show that $f$ has exactly $5$ real roots. (Since $f$ has negative discriminant, $f$ has either exactly $1$ or exactly $5$ real roots, hence it suffices to show that $f$ has at least $2$ real roots.)

What kind of permutations do (1), (2) (separately) imply occur in $\operatorname{Gal}(f)$?

1. So, $\operatorname{Gal}(f)$ contains a $7$-cycle.

2. Complex conjugation exchanges the $2$ nonreal roots of $f$, hence it determines a transposition in $\operatorname{Gal}(f)$.