Question: Suppose $\zeta$ is a root of unity and $f\in \mathbb{Q}[X]$. Show that $f(\zeta)\neq2^{1/4}$.
This is from the UCLA fall '16 algebra qual. So far I haven't gotten far besides my initial observation that if we suppose that $f(\zeta)=2^{1/4}$, then as $\mathbb{Q}(\zeta)/\mathbb{Q}$ is Galois, we get $g(X)=X^4-2$ splits in $\mathbb{Q}$ and so we have $\mathbb{Q}(\zeta)/E/\mathbb{Q}$ where $E$ is the splitting field of $g$. Since $|E:\mathbb{Q}|=2^3$ we get that $2^3|\varphi(ord(\zeta))$ and this eliminates a lot of possible roots of unities. However, proving the general result escapes me.
Each root of unity $\zeta$ generates a cyclotomic extension $\Bbb Q(\zeta)/\Bbb Q$ with Abelian Galois group. Its subfields are all Galois over $\Bbb Q$. But $\Bbb Q(2^{1/4})/\Bbb Q$ is not a Galois extension.