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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 $F(\zeta_n)$.

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Yes, this is always possible. First note that the automorphism of $F$ induces and injective field homomorphism $F\to F(\zeta_{n})$. Then write $F(\zeta_{n})\cong F[T]/(f)$, where $f$ is the minimal polynomial of $\zeta_{n}$. By the universal property of the polynomial ring and of the quotient, sending the class of the variable $T$ to any root of $f$ in the right hand side gives you the desired automorphism of $F(\zeta_{n})$. The result is bijective because it is an injective homomorphism of $F$-vector spaces of the same finite dimension.

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