Let $n \geqslant 1$ and $m \geqslant 1$ be two integers and $\alpha _ { 1 } , \ldots , \alpha _ { n } , \beta _ { 1 } , \ldots , \beta _ { m }$ be complex numbers. We define: $$\begin{aligned}
& A ( X ) = \left( X - \alpha _ { 1 } \right) \left( X - \alpha _ { 2 } \right) \cdots \left( X - \alpha _ { n } \right) \\
& B ( X ) = \left( X - \beta _ { 1 } \right) \left( X - \beta _ { 2 } \right) \cdots \left( X - \beta _ { m } \right)
\end{aligned}$$ Show that if $A ( X )$ and $B ( X )$ have rational coefficients, then the polynomials $$\prod _ { i = 1 } ^ { n } \prod _ { j = 1 } ^ { m } \left( X - \alpha _ { i } \beta _ { j } \right) \text { and } \prod _ { i = 1 } ^ { n } \prod _ { j = 1 } ^ { m } \left( X - \alpha _ { i } - \beta _ { j } \right)$$ also have rational coefficients.
Let $n \geqslant 1$ and $m \geqslant 1$ be two integers and $\alpha _ { 1 } , \ldots , \alpha _ { n } , \beta _ { 1 } , \ldots , \beta _ { m }$ be complex numbers. We define:
$$\begin{aligned}
& A ( X ) = \left( X - \alpha _ { 1 } \right) \left( X - \alpha _ { 2 } \right) \cdots \left( X - \alpha _ { n } \right) \\
& B ( X ) = \left( X - \beta _ { 1 } \right) \left( X - \beta _ { 2 } \right) \cdots \left( X - \beta _ { m } \right)
\end{aligned}$$
Show that if $A ( X )$ and $B ( X )$ have rational coefficients, then the polynomials
$$\prod _ { i = 1 } ^ { n } \prod _ { j = 1 } ^ { m } \left( X - \alpha _ { i } \beta _ { j } \right) \text { and } \prod _ { i = 1 } ^ { n } \prod _ { j = 1 } ^ { m } \left( X - \alpha _ { i } - \beta _ { j } \right)$$
also have rational coefficients.