Show that the polynomial $f(X)=X^{5}+27 X+16$ has no rational roots. Show that the splitting field of $f$ over the finite field $\mathbb{F}_{3}$ is an extension of degree 4 . Hence deduce that $f$ is irreducible over the rationals. Prove that $f$ has precisely two (non-multiple) roots over the finite field $\mathbb{F}_{7}$. Find the Galois group of $f$ over the rationals.

[You may assume any general results you need including the fact that $A_{5}$ is the only index 2 subgroup of $S_{5}$.]