For any prime $p \neq 5$, explain briefly why the Galois group of $X^{5}-1$ over $\mathbb{F}_{p}$ is cyclic of order $d$, where $d=1$ if $p \equiv 1 \bmod 5, d=4$ if $p \equiv 2,3 \bmod 5$, and $d=2$ if $p \equiv 4$ $\bmod 5 .$

Show that the splitting field of $X^{5}-5$ over $\mathbb{Q}$ is an extension of degree 20 .

For any prime $p \neq 5$, prove that $X^{5}-5 \in \mathbb{F}_{p}[X]$ does not have an irreducible cubic as a factor. For $p \equiv 2$ or $3 \bmod 5$, show that $X^{5}-5$ is the product of a linear factor and an irreducible quartic over $\mathbb{F}_{p}$. For $p \equiv 1 \bmod 5$, show that either $X^{5}-5$ is irreducible over $\mathbb{F}_{p}$ or it splits completely.

[You may assume the reduction mod p criterion for finding cycle types in the Galois group of a monic polynomial over $\mathbb{Z}$ and standard facts about finite fields.]