# Extension of automorphism of field

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 $$F(\zeta_n)$$.

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.