Suppose $K_1,K_2$ are normal extensions over a field $F$ . Suppose the set of minimal polynomials of elements in $K_1$ over $F$ = the set of minimal polynomials of elements in $K_2$ over $F$ . Then are $K_1,K_2$ isomorphic ?
My attempt:
I have proved the fact that an algebraic extension $K |F$ is normal iff $\text{min}_F (\alpha)$ splits in $K[X] , \forall \alpha \in K$ . Combining this with the hypothesis that the set of minimal polynomials of elements in $K_1$ over $F$ = the set of minimal polynomials of elements in $K_2$ over $F$, it seems to me that in fact $K_1$ must equal $K_2 $, but I do not have proof.
Thanks in advance for the help!