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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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Hint (not really a hint as it pretty much gives you the answer): If $f(x)$ is the minimum polynomial of $\alpha, \beta$ then both $K(\alpha), K(\beta)$ are isomorphic to $K[x]/(f(x))$.

If your isomorphism fixes $K$ (i.e. acts as identity on $K$) then they have the same minimal polynomial.

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  • $\begingroup$ Cheers, now that I see it it is very clear. $\endgroup$ – GiulioB Nov 28 '12 at 17:40
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The last part of the question is not clear. First, extensions do not contain polynomials. Second, if you are asking, given that $K(\alpha)$ and $K(\beta)$ are isomorphic as extensions of $K$, is it true that $\alpha$ and $\beta$ have the same minimum polynomial, then certainly the answer is no. E.g., ${\bf Q}(\root4\of2)$ and ${\bf Q}(i\root4\of2+17)$ are isomorphic extensions of the rationals, but $\root4\of2$ and $i\root4\of2+17$ don't have the same minimum polynomial. But perhaps what you are asking is, given that $K(\alpha)$ and $K(\beta)$ are isomorphic as extensions of $K$, is there an element $\gamma$ such that $K(\beta)=K(\gamma)$ and $\alpha$ and $\gamma$ have the same minimum polynomial, then Rankeya's answer applies; take $\gamma$ to be the image of $\alpha$ under the isomorphism.

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