this is my problem:
Suppose $K|F$ is a normal extension. Prove that for every $\alpha ,\beta \in K$ that have the same minimal polynomial over $F$ there is a $F$-algebra automorphism of $K$ (automorphism which is identity over $F$) $\phi:K\rightarrow K$ such that $\phi(\alpha)=\beta$.
we know that if $\alpha ,\beta \in K$ have the same minimal polynomial over $F$ then $F(\alpha)\simeq_F F(\beta)$,but how can i continue?
any hint is welcomed!
thank you very much!