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In van der Waerden's Algebra, a separable element with respect to a field $F$ is defined as the following: If $\alpha$ is a root of a polynomial irreducible in $F[X]$ which has only separated(simple) roots, then $\alpha$ is called separable.

Looking at the definition, I don't find it clear how and why some elements are not separable. Could someone give an example of a field and an inseparable element over it?

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The element $t$ (transcendental over $K$) in the field extension $K(t)/K(t^p)$ is not separable, when $K$ has characteristic $p$.

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  • $\begingroup$ Can you please provide more details? $\endgroup$ – odnerpmocon Feb 5 '15 at 16:48
  • $\begingroup$ The minimal polynomial of $t$ is $X^p-t^p$, which is irreducible over $K(t^p)$ due to Eisenstein. Now you should check that this polynomial is obviously not separable. $\endgroup$ – MooS Feb 5 '15 at 16:51
  • $\begingroup$ I see that $X^p-t^p=(X-t)^p$, so it's not separable. But can you explain how you obtained the minimal polynomial? $\endgroup$ – odnerpmocon Feb 5 '15 at 16:56
  • $\begingroup$ Well it is the obvious choice, isn't it? We just have to check irreducibility, which i already did. $\endgroup$ – MooS Feb 5 '15 at 16:57

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