Let $K$ be a field and $f$ a separable polynomial over $K$ of degree $n$. Explain what is meant by the Galois group $G$ of $f$ over $K$. Show that $G$ is a transitive subgroup of $S_{n}$ if and only if $f$ is irreducible. Deduce that if $n$ is prime, then $f$ is irreducible if and only if $G$ contains an $n$-cycle.

Let $f$ be a polynomial with integer coefficients, and $p$ a prime such that $\bar{f}$, the reduction of $f$ modulo $p$, is separable. State a theorem relating the Galois group of $f$ over $\mathbb{Q}$ to that of $\bar{f}$ over $\mathbb{F}_{p}$.

Determine the Galois group of the polynomial $x^{5}-15 x-3$ over $\mathbb{Q}$.