I'm trying to give an example of a normal field extension $K|F$ that is not separable. I now that if $F$ is finite or char$(F)=0$, $K|F$ is automatically separable, thus, I must look into infinite fields with char$(F)=p>0$.

The only fields I know that satisfy those properties are the rational function field over the $\mathbb{Z}_p$'s, but the examples I've seen before with those fields are never trivial! I have an idea but I'm not sure if it's right:

Let $p$ be a prime, $F=\mathbb{F}_p(t^p)$ and $K=F(t)$. $K|F$ is not separable since the minimal polynomial $\mbox{min}(F,t)=x^p-t^p$ has repeated roots. However, this extension is normal since $K$ is the splitting field of $t$ over $F$.

I hope someone can make a comment about this. Thanks.

  • $\begingroup$ You can also say that $K|F$ is normal because there is only one $F$-homomorfism $\alpha$ from $K$ into an algebraic closure $\bar{F}$ of $F$: $\alpha(t)$ is the unique element $a$ of $\bar{F}$ such that $a^p=t^p$. $\endgroup$ – Diego Oct 20 '14 at 16:16
  • $\begingroup$ Why $\alpha(t)$ is unique element of F closure? $\endgroup$ – Ripan Saha Oct 21 '14 at 4:44

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