Prove that the Galois group $G$ of the polynomial $X^{6}+3$ over $\mathbf{Q}$ is of order 6 . By explicitly describing the elements of $G$, show that they have orders 1,2 or 3 . Hence deduce that $G$ is isomorphic to $S_{3}$.

Why does it follow that $X^{6}+3$ is reducible over the finite field $\mathbf{F}_{p}$, for all primes $p ?$