Originally Posted by mathmari
In this the chapter 3.4 (p.11) is about separability. This proposition is the 3.4.5 and the only example of an inseparable polynomial that I found is the one that is at 3.4.8.

This gives example 1.2 with $x^3-2$ in $F_3[x]$ as an inseparable polynomial.
But its definition of inseparable seems to be different from your definition.
Btw, seems to give a different definition as well.

I can see that your book considers $x^3-2$ reducible, since it is reducible over $F_3$. It seems that my book reference considers $x^3-2$ irreducible, but to be honest I'm not sure why. Wiki's definition of seems to imply that it's reducible.