Question about what is a separable polynomial Here is the definition of separable polynomial as I understand it.
Let $K$ be a splitting field for a polynomial $f(x)\in F[x]$. If $f(x)$ is irreducible, then $f(x)$ is separable if the roots in $K$ all have multiplicity $1$. If $f(x)$ is reducible, then $f(x)$ is separable if all the irreducible factors of $f(x)$ are separable.
My question is whether the irreducible factors can share roots. For example, if $g(x)$ is a separable irreducible polynomial, then if $g(x)^2$ separable?

There seems in fact two definitions of separable polynomial.
Using the note(Fields and Galois Theory
J.S. Milne
), in page 33:

Definition 2.22 A polynomial is separable if it is nonzero and satisfies the equivalent conditions on (2.21).

conditions on (2.21) is that the polynomial has only simple roots.
But there is also a footnote on this definition (in page 33 too):

This is Bourbaki’s definition. Often, for example, in the books of Jacobson and in earlier versions of
these notes, a polynomial  is said to be separable if each of its irreducible factors has only simple roots.

