Isomorphism between algebraic extensions and irreducible polynomials

Question

I need to prove wether the following statement is true or false, but i have no idea on how to solve it:

Being $E/K$ an algebraic field extension and $\alpha, \beta \in E$ algebraics elements over $K$. If there's a field isomorfism $\phi:K(\alpha)\to K(\beta)$ so that $\phi(k)=k \enspace \forall k\in K$ $\Rightarrow \enspace \exists p(x)\in K[x]$ irreducible so that $p(\alpha)=p(\beta)=0$.

Any ideas? Thanks a lot!

Answer 1

No. Take $K = \mathbb{Q}$ and $\alpha = 0, \beta =1$, $\phi= id_{\mathbb{Q}}$. Any polynomial $p$ over $\mathbb{Q}$ that vanishes in $0$ and $1$ contains the factors $X, X-1$. I.e.

$$p= X(X-1) A(X)$$

and clearly $p$ is not irreducible.