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整系数多项式$f(x)=\prod_{i=1}^n(x-\alpha_i),\alpha_i\in\Bbb C$与$g(x)=\prod_{j=1}^m(x-\beta_j),\beta_j\in\Bbb C$
多项式$$h(x)=\operatorname{Resultant}(f(t),g(x-t),t)=\prod_{i=1}^n\prod_{j=1}^m(x-(\alpha_i+\beta_j))$$可以表示为$x$的整系数多项式排成的行列式。
因此 $h(x)$ 的系数都是整数。
来源: 代数整数之和为代数整数
MathWorld
Given $p$ and $q$, then
$$h(x)=\rho(q(t),p(x-t))
\tag7$$
is a polynomial of degree $mn$, having as its roots all sums of the form $\alpha_i+\beta_j$. |
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