Galois Theory: when does [E : F] \= |Gal(E/F)| ?
I'm trying to find a field F and a polynomial with splitting field E such that . 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 (the field of rational functions with elements of 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 or . Any help or hints would be much appreciated.