= Solution
Factorization over the <rational numbers> gives
$$
f(x)=(x-1)(x+1)(x^3-7).
$$
The rational roots add nothing to the <splitting field>. By <Eisenstein's criterion> at $7$, $x^3-7$ is irreducible. Put $a=\sqrt[3]7$ and $\zeta=e^{2\pi i/3}$. Its <splitting field> is $L=\mathbb Q(a,\zeta)$. The real <field> $\mathbb Q(a)$ has degree $3$; the nonreal element $\zeta$ has degree $2$ over it. Thus $[L:\mathbb Q]=6$, and its faithful action on the three nonrational roots identifies
$$
\boxed{\operatorname{Gal}(f)\cong S_3.}
$$
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