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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?