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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.

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1 Answer 1

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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 $\beta=1+\sqrt2$.

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  • $\begingroup$ Thanks for your answer. Isn't there more connection between $\alpha$ and $\beta$? $\endgroup$
    – Mary
    Apr 15, 2021 at 10:20
  • $\begingroup$ @Mary, of course, $\beta=f(\alpha)$ and $\alpha=g(\beta)$ with $f,g \in K[x]$. $\endgroup$
    – lhf
    Apr 15, 2021 at 10:26
  • $\begingroup$ How about when $K$ is of prime characteristic? Can we say anything more? $\endgroup$
    – Mary
    Apr 15, 2021 at 10:28

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