Consider polynomials $f,g \in k[x]$ of positive degrees $m,n$ respectively. Let $I$ denote the ideal in $k[x]$ generated by $f$, and let $\mu$ denote the multiplication map
$$\mu:k[x]/I \to k[x]/I,h+I \mapsto gh+I$$
Demonstrate that $res_x(f,g)=LC(f)^{\deg(g)}\det(\mu)$.
Now the resultant is the determinant of the corresponding Sylvester matrix:
$\begin{pmatrix} f_m & f_{m-1} & \cdots & & f_0 & & & \\ & f_m & f_{m-1} & \cdots & & f_0 & & \\ & & \ddots & & & & \ddots & \\ & & & f_m & f_{m-1} &\cdots & & f_0 \\ g_n & g_{n-1} & \cdots & & g_0 & & & \\ & g_n & g_{n-1} & \cdots && g_0 & & \\ & & \ddots & & & & \ddots & \\ & & & g_n & g_{n-1} & \cdots & & g_0 \\ \end{pmatrix}$
The determinant can be calculated using Laplace expansion. If we start at $f_m=LC(f)$ then we can do a Laplace expansion exactly $\deg(g)$ times (along the diagonal). However if we do the Laplace expansion always for the first column of the new submatrix then we also have the $g_n$'s in the new submatrices?
Also I struggle to understand what the determinant of a multiplication map is. I've read that it involves something with bases but I don't really understand that.
Any help would be appreciated!