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1) I know K/F is normal if K is a splitting field of some S subset F.

2) K/F is a separable extension if every element of K is separable.

...but am having problems coming up with an extension that can be neither (not 1 and not 2)

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How about $L=F_5(\root{15}\of t)$ over $K=F_5(t)$? There's an intermediate field $E=F_5(\root5\of t)$ which is inseparable over $K$, and $L$ is not normal over $E$.

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