In my class notes, I have two theorems which don't quite seem to fit together. Maybe you can help me out. Thm 1 If p(x) in F[x] splits in K, then E=F(a1,...,an) is the splitting field of p(x) in K (the a_i's are the roots of p(x)). Thm 2 If p(x) in F[x], then the splitting field of p(x) is unique up to isomorphism. I'm clearly missing something big here. Doesn't (1) imply (2)? Isn't (1) even stronger than (2)? What's an example of a polynomial with two distinct but isomorphic splitting fields?