$\mathbb{k}$ is a field. $\alpha \in \mathbb{k}$. Prove that if polynomial $f (x)=x^{n} - 1$ has all the roots over $\mathbb{k}$, then $g(x) = x^{n} - \alpha$ is either irreducible over $\mathbb{k}$ or $\exists d>1, d|n$, that $x^{d} - \alpha$ has at least one root over $\mathbb{k}$
So there exist $n$ elements of $\mathbb{k}$ of order $n$. And they are closed under multiplication $(ab)^{n} = a^{n}b^{n} = 1$. We have $1^{n} = 1$. And $(a^{-1})^{n} = (a^{n})^{-1} = 1$. So it's a group under multiplication. I don't know if it helps in solving this problem, but that's all that i've figured out. What i should do with $x^{n} - \alpha$?
