The following is taken from Problem 6.2 on p.86 of The Theory of Algebraic Numbers by Harry Pollard and Harold G. Diamond (Dover edition).
Consider the quartic polynomial $$(x-am)\,(x-an)\,(x-bm)\,(x-bn)\ \equiv\ x^4-c_1x^3+c_2x^2-c_3x+c_4$$
It is not stated what the $a,b,m,n$ are – they can be integers, rational numbers, etc. The question simply asks to verify that the coefficients $c_i$ can be expressed as the following product: $$c_i(a,b,m,n)\ =\ p_i(a,b)\cdot q_i(m,n)\quad(i=1,2,3,4)$$
where $p_i(a,b)$ is an expression in $a,b$ only and $q_i(m,n)$ is an expression in $m,n$ only. This is easily done for $c_1,c_3,c_4$ but I have a problem with $c_2$ (the coefficient of $x^2$). How can I write $c_2=p_2q_2$ where $p_2$ involves $a,b$ only and $q_2$ involves $m,n$ only?
Note that $p_2$ and $q_2$ need not be integers even if $a,b,m,n$ themselves are. If $a,b,m,n$ are integers, then $c_2$ is necessarily an integer, but $p_2,q_2$ themselves need not be – in fact, they need not even be rational.
I would appreciate some help here. Thanks.
\begin{align*} c_{1} &= \alpha+\beta+\gamma+\delta \\ &= am+an+bm+bn \\ &= (a+b)(m+n) \\ c_{2} &= \alpha \beta+\alpha \gamma+\alpha \delta+ \beta \gamma+\beta \delta+\gamma \delta \\ &= (am)(an)+(am)(bm)+(am)(bn)+(an)(bm)+(an)(bn)+(bm)(bn) \\ &= (a^2+b^2)mn+ab(m^2+n^2)+2abmn \\ c_{3} &= \alpha \beta \gamma+\alpha \beta \delta+ \alpha \gamma \delta+\beta \gamma \delta \\ &= (am)(an)(bm)+(am)(an)(bn)+(am)(bm)(bn)+(an)(bm)(bn) \\ &= ab(a+b)mn(m+n) \\ c_{4} &= \alpha \beta \gamma \delta \\ &= a^2 b^2 m^2 n^2 \end{align*}