The background of this question comes from Pinter's Abstract Algebra, Chapter 27, Exercise J.
Suppose $a(x) \in F[x]$, and $K$ is an extension of $F$. An element $c\in K$ is called a multiple root of $a(x)$ if $(x-c)^m|a(x)$ for some $m > 1$.
If $x-c$ is a common factor of $a(x)$ and $a'(x)$, then $a(x)$ and $a'(x)$ have a common factor of degree $>1$ in $F[x]$.
Let $a(x) = (x-c)^2q(x) \in K[x]$. Then $a'(x) = 2(x-c)q(x) + (x-c)^2q'(x)$. Since both $a(x)$ and $a'(x)$ have a common root $c$ in some extension of $F$, they have a common factor $p(x)$ of positive degree in $F[x]$.
Obviously, $\operatorname{deg} p(x) > 1$ if $c\in K$ and $c\not\in F$. However, it seems $\operatorname{deg} p(x)\ge 1$ if $c\in F$.
For example, let $a(x)=(x-1)^2\in\Bbb{Q}[x]$, so $a'(x)=2(x-1)$, and $(x-1)^2|a(x)$. Their common factor is therefore $x-1$ with degree $1$.
Correct?
