Can different subfields have the same Minimal Polynomial? I am trying to see if different subfields (of $F(\alpha)$ containing $F$) can have the same minimal polynomial?
I'm hoping that the answer is no, so that if two subfields do have the same minimal polynomial, then they must be the same subfield!
Edited to add in the whole question:
Let $F(\alpha) : F$ be a simple algebraic extension.  Let $E_1$ and $E_2$ be intermediate fields (that is, subfields of $F(\alpha)$ containg $F$).  Prove that if $\alpha$ has the same minimal polynomial over $E_1$ and $E_2$, then $E_1 = E_2$.
 A: Your question is not clear, since subfields do not have minimal polynomials. I believe that your question is something along these lines:
"Given $F(\alpha)$ and $F(\beta)$, two algebraic extension fields of $F$, and knowing that the minimal polynomial of $\alpha$ over $F$ is the same as the minimal polynomial of $\beta$ over $F$, can I conclude that $F(\alpha)=F(\beta)$?"
I believe that the answer to this question is no. Consider the following polynomial:
$$x^3-2=(x-\sqrt[3]2)(x-\sqrt[3]2\omega)(x-\sqrt[3]2\omega^2)$$
where $\omega$ in the third root of unity. Notice that none of the roots belong to $\mathbb{Q}$. Also this polynomial is irreducible in $Q[X]$ by Eisenstein criterion. Being monic, it is a minimal polynomial of $\sqrt[3]2$ and $\sqrt[3]2\omega.$ But $\mathbb{Q}[\sqrt[3]2] \neq \mathbb{Q}[\sqrt[3]2\omega]$, since one field extension contains complex numbers and the other one does not. 
So,the extensions are not the same. But are they isomorphic? What do you think?
A: Yes, they can.  Example: Let p(x) be an irreducible cubic over $F$ with three roots $r_1$, $r_2$, $r_3$.  The three fields $F(r_1)$, $F(r_3)$, $F(r_3)$ will be distinct, and p(x) will be the minimal polynomial for all of them.  And they will all be common subfields of the spliting field of p(x) [$F(r_1,r_2,r_3)$, which is also representable as $F(\alpha)$ for some $\alpha$].
[note: this is a slight generalization of the answer by @Paquarian]
