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.
Hi zoek,
This is confusing to me because I have a theorem in my textbook (Rotman, "Galois Theory") which says:
If is a separable polynomial and if is its splitting field, then .
I also have a theorem which says every field of characteristic 0 is perfect i.e. every polynomial is separable. How does this make sense in light of your example? How did you get that and ? It seems to me that a splitting field for your polynomial would contain both the real cube root of 2 and the two imaginary cube roots of 2, and so .
Thanks for the help.
one thing that occurs to me is to set K = Z3(u), rational functions in u over Z3, and set f(x) = x^3 - u in K[x]. i believe f is irreducible over K, but if α is a cube root of u, then
f(x) = (x - α)^3. so E = K(α) is a splitting field of f over K, and [E:K] = 3. since any automorphism of E has to send α-->α, Gal(E/K) = {1}.