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K is a field and $char(K)=p$, $K(s,t)$ is the field of rational functions in variable $s$ and $t$. Now $K(s^{p},t^{p})\subseteq K(s,t)$. Prove that $K(s,t)$ is a simple field extension of $K(s^{p},t^{p})$ ( which means $K(s,t)=K(s^{p},t^{p})(u)$ ).

Right now I can prove that the degree of this extension is $p^{2}$ and I want to prove by contradiction. But I don't know how to derive a contradiction.

Jack
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  • This seems very close to you are asking: https://math.stackexchange.com/questions/276627/a-finite-field-extension-that-is-not-simple – user404127 Apr 26 '17 at 00:41
  • I think you want to prove that the extension is not simple. – user26857 Apr 26 '17 at 22:30
  • On the contrary, it can be shown that this extension is not simple, and this is the most classical example of non simple extension. – MikeTeX Apr 27 '17 at 18:09

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