TITLE: Separable polynomial definition (Confused)
QUESTION [5 upvotes]: I understand there is a new and old definition of separable polynomial (https://en.wikipedia.org/wiki/Separable_polynomial).
For the following definition:
a) A polynomial over $F$ is said to be separable if it has no multiple roots. (given in Dummit & Foote)
b) a polynomial P(X) over a given field K is separable if its roots are distinct in an algebraic closure of K (Wikipedia)
c) A polynomial f in F[X] is a separable polynomial if and only if every irreducible factor of f in F[X] has distinct roots (https://en.wikipedia.org/wiki/Separable_extension#cite_note-6)
Question) Which is the correct definition?
I can see that (b) implies (c), for instance, but not sure if they are equivalent.
Thanks for help.

REPLY [7 votes]: 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.