Let $F$ be a field of characteristic zero, $\overline{F}$ be the algebraic closure of $F$. Let $\zeta_n$ be a primitive $n$-th root of unity in $\overline{F}$. Then it is well-known that $F(\zeta_n)$ is a finite Galois extension of $F$.
Q. If $\sigma:F\rightarrow F$ is a field automorphism, then is it always possible to extend it to an automorphism of $F(\zeta_n)$?
The question might be trivial,I do not know. But usually, in Galois theory, I had visited most of the time extension of identity automoorphism of a field to its finite (or even Galois) extensions. Here I am considering the problem of extending any automorphism of $F$ to an automorphism of $F(\zeta_n)$.