Suppose $L$ is a splitting field of $F$ and let $f(x)$ be an separable irreducible polynomial in $F[x]$. Let $\sigma \in Aut(L/F)$ then we know that if $\alpha$ is a root of $f(x)$ then $\sigma(\alpha)$ is also a root of $f(x)$. Does it imply that degree of minimal polynomial of $\alpha$ is same as degree of minimal polynomial of $\sigma(\alpha)$?
Or $L$ is required to be Galois Extension of $F$?