How can I see that the resultant of two polynomials can be expressed by their coefficients?

Question

I define the resultant of $f_1, f_2$ by $\prod (x_i-y_j)$ where $x_i, y_j$ are the roots of $f_1, f_2$. Then how can I see this can be expressed by the coefficients of $f_1,f_2$?

Answer 1

Let $g=\prod_{i,j} (x_i-y_j)$. Work over a field $k$, and also assume that $f_1,f_2$ are monic.

$g$ is a polynomial combination of the $x_i$ with coefficients in $k(\{y_j\})$. Clearly, it is symmetric, i.e. permuting the $x_i$ does not change $g$.

By the fundamental theorem of symmetric polynomials, this implies that $g$ is a polynomial combination of elementary symmetric polynomials $e_i$ of the $x_i$, which are just $\pm$ the coefficients of $f_1$.

By the same argument, applied to $g$ with coefficients in $k(e_i)$, we can write $g$ as a polynomial combination of the coefficients of $f_1$ and the coefficients of $f_2$.