Let $k$ be a field. For $m$ a positive integer, consider $X^{m}-1 \in k[X]$, where either char $k=0$, or char $k=p$ with $p$ not dividing $m$; explain why the polynomial has distinct roots in a splitting field.

For $m$ a positive integer, define the $m$ th cyclotomic polynomial $\Phi_{m} \in \mathbb{C}[X]$ and show that it is a monic polynomial in $\mathbb{Z}[X]$. Prove that $\Phi_{m}$ is irreducible over $\mathbb{Q}$ for all $m$. [Hint: If $\Phi_{m}=f g$, with $f, g \in \mathbb{Z}[X]$ and $f$ monic irreducible with $0<\operatorname{deg} f<\operatorname{deg} \Phi_{m}$, and $\varepsilon$ is a root of $f$, show first that $\varepsilon^{p}$ is a root of $f$ for any prime $p$ not dividing $m$.]

Let $F=X^{8}+X^{7}-X^{5}-X^{4}-X^{3}+X+1 \in \mathbb{Z}[X]$; by considering the product $\left(X^{2}-X+1\right) F$, or otherwise, show that $F$ is irreducible over $\mathbb{Q}$.