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I am trying to construct a finite field extension $L/K$ of prime characteristic $p$, such that $\gcd([L:K],p)=1$, and that has an inseparable element.

Trying to construct an example of such extension, I could only find infinite extensions and extensions where $\gcd([L:K],p)\neq1$, but I can neither prove that the mentioned conditions define a separable extension.

please help :)

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  • $\begingroup$ If you can show the composition of fields as a separable and purely inseparable extension, you'll see such a thing is impossible. $\endgroup$ – Adam Hughes Oct 5 '15 at 17:15
  • $\begingroup$ I am not sure I got it... $\endgroup$ – Daniel Oct 5 '15 at 17:21
  • $\begingroup$ Purely inseparable extensions are of the form $K(x)$ where $x^{p^n}\in K$ for some $n$. $\endgroup$ – Adam Hughes Oct 5 '15 at 19:41

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