Are all finite Extensions of Fp (the field w/ p elements) Galois?

My understanding is no, since finite Galois $\displaystyle \iff$ normal and separable.

Then as a counterexample, in $\displaystyle \frac{F_3}{<x^2+x-1>}$, x^2+x-1 has one root (x or equivalently x+<x^2+x-1>) but no others (right?) so it is not normal and thus not Galois.