(a) Let $A , B \in \mathbb { Q } [ X ]$ be two polynomials that have a common root in $\mathbb { C }$. Show that $A$ and $B$ are not coprime in $\mathbb { Q } [ X ]$. (b) Show that the roots of $\Pi _ { \alpha }$ in $\mathbb { C }$ are simple.
(a) Let $A , B \in \mathbb { Q } [ X ]$ be two polynomials that have a common root in $\mathbb { C }$. Show that $A$ and $B$ are not coprime in $\mathbb { Q } [ X ]$.\\
(b) Show that the roots of $\Pi _ { \alpha }$ in $\mathbb { C }$ are simple.