Galois Theory: when does [E : F] \= |Gal(E/F)| ?

I'm trying to find a field F and a polynomial $\displaystyle f(x) \in F[x]$ with splitting field E such that $\displaystyle [E : F] \not = |Gal(E/F)|$. I know that f(x) must be inseparable for this to work, and so F must be infinite with characteristic p > 0. But apart from that, I'm stumped. The fields $\displaystyle \mathbb{Z}_p(x)$ (the field of rational functions with elements of $\displaystyle \mathbb{Z}_p$ as coefficients) for some prime p are the only such fields I can think of, and I can't think of an example in that field that makes it easy for me to compute $\displaystyle [E : F]$ or $\displaystyle |Gal(E/F)|$. Any help or hints would be much appreciated.