Why is it true that any polynomial $f(x)$ with coefficients in a finite field $F_q$ is separable over that field? Why is it true that any polynomial $f(x)$ with coefficients in a finite field $F_q$ is separable over that field?
I'm confused because the polynomial $x^2+1$ in $F_2[x]$ is inseparable ($f(x)$ and $f'(x)$ are not coprime), and yet apparently this property is true.  Am I missing something? 

The definition of a field $E$ being perfect is that every irreducible polynomial $p\in E[x]$ is separable (Wikipedia link). But the polynomial $x^2+1$ is equal to $(x+1)^2$ in $\mathbf{F}_2[x]$, so it's not irreducible. 
Similarly, the field $\mathbf{Q}$ being perfect doesn't conflict with $x^2+2x+1=(x+1)^2\in\mathbf{Q}[x]$ having repeated roots.
