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I'm looking for an example of an normal extension but not separable; all I know is that $\mathbb{F}_p(t)$ is not separable since $X^p-t$ is not. Thank you for your time

Houda
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1 Answers1

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Recall (or convince yourself) that every extension of degree $2$ is normal and use the example you recalled with a suitable choice of $p$.

quid
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    yes, I found it ! for $K=\mathbb{F}_2(\sqrt{t})$ which is a quadratic extension over $\mathbb{F}_2(t)$ so it's normal as you said (I still have to check it out), the polynomial I gave above is not separable. thanks – Houda Dec 11 '14 at 00:25
  • Why is $X^2-t$ not separable? It has only simple roots ($\sqrt{t}$ and $- \sqrt{t}$)... – rmdmc89 Oct 05 '16 at 20:57
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    @AguirreK It is in characteristic $2$ the "roots" you give are actually one root (with multiplicity two). – quid Oct 05 '16 at 21:00
  • You are right, @quid, thank you – rmdmc89 Oct 05 '16 at 21:03