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.