Separable polynomial definition (Confused)

Solution 1:

The definitions are not equivalent. a) and b) is a stronger condition than c). The question is whether you want something like $x^2$ to be separable. For a),b) $x^2$ is not separable, for c) $x^2$ is separable, since its irreducible factors are separable.

Definition c) is in my opinion the better definition since it makes sure that a polynomial $f \in K[X]$ is separable if and only if the splitting field is a separable extension of $K$.

Furthermore, with definition c), you have that a field is perfect if and only if every polynomial over that field is separable.

With definitions a),b) you have to restrict to irreducible polynomials.