Let $\alpha=2^{1/5}$ ,$\zeta=e^{2i\pi/5}$ and $K=\mathbb{Q}(\alpha\zeta)$, can the field automorphism of K be extended to an automorphism of C? Let $\alpha=2^{1/5} $ and $\zeta=e^{2i\pi/5}$.Let $K=\mathbb{Q}(\alpha\zeta) $ . I just read a theorem which is $-$ Any automorphism of a subfield of $\mathbb{C} $ can be extended to an automorphism of $\mathbb{C}$. Per my previous Question the only automorphism of $K$ is identity automorphism, and by the above theorem it can be extended to an automorphism of $\mathbb {C} $ . My question is $-$ Is the extended automorphism identity or nonidentity automorphism of $\mathbb{C} $. Edit: Here the question is whether the extended automorphism is identity or non-identity automorphism of $\mathbb{C}$. I am sure I didn't ask THIS question anywhere else.

Let $L={\bf Q}(\alpha,\zeta)$, so $L$ is a Galois extension of $\bf Q$ containing $K$. There are twenty automorphisms of $L$ fixing $\bf Q$. Many of them also fix $K$. Take one of those, not the identity, and extend it to an automorphism of $\bf C$, and you're done.