Question about what is a separable polynomial

Question

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?

Answer 1

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.

Answer 2

Your stated definition of a separable polynomial is not the actual definition. A separable polynomial $f(x) \in F[x]$, $F$ a field, is one that has no repeated roots in an algebraic closure of $F$. So, if $f(x)$ has a repeated irreducible factor $g(x)$, it is not separable even if $g(x)$ is.