Let K be a subfield of C, the field of complex numbers, and f an irreducible polynomial in K[X]. Then f and Df are coprime so there exist a,b in K[X] such that af + bDf = 1 (D is the formal derivative operator). Now what I don't understand is why this equation implies f and Df are coprime when viewed over C[X]. Doesn't f split into linear factors over C[X] so is it not possible that a factor divides Df (not always but could happen?). Although if I write f in linear factors and then compute Df I can see that no linear factor of f divides it, but it does if one of the linear factors is repeated?
Certainly if there exist a and b in K[X] such that af + bDf = 1, then there exist a and b in C[X] such that af + bDf = 1. This is equivalent to saying that none of the linear factors of f in C[X] are repeated (if there was such a linear factor, it would divide both f and Df).