Simple algebraic field extensions of prime degree

Question

Let $K(\alpha)/K$ be a simple algebraic field extension of prime degree $p$. Suppose $\beta \in K(\alpha)$ with $\beta\not\in K$ and $\beta\not=\alpha$. What can we say about $\beta$? Is it necessarily a $K$-conjugate of $\alpha$? Thanks for any help.

Answer 1

We can only say that $K(\beta)=K(\alpha)$.

$\beta$ is not necessarily a $K$-conjugate of $\alpha$. Consider for instance $\alpha=\sqrt 2$ and $\beta=1+\sqrt2$.