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Let $\alpha$ and $\beta$ be algebraic elements of an extension $L$ of $K$. Is it always true that if they have the same minimum polyomial then $K(\alpha)$ is isomorphic to $K(\beta)$?

I think it is true, it follows pretty much straightforward out of the definition of miminum polynomial. My question is: are there exceptions? If not, how can you prove it?

And also: does it work the other way round, meaning if two extensions are isomorphic then they always contain such a minimum polynomial?

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