Solution
= Solution
Over the <finite field> $\mathbb F_7$,
$$
x^3+x^2+x+1=(x+1)(x^2+1).
$$
The nonzero squares modulo $7$ are $1,2,4$, so $-1$ is not a square. Adjoining a root of $x^2+1$ produces $\mathbb F_{49}$ and already gives both roots. Its <Frobenius automorphism> has order $2$ and interchanges them, whence
$$
\boxed{\operatorname{Gal}(f)\cong C_2.}
$$