The formula $R(fg,h)=R(f,h)R(g,h)$ follows easily if you express each resultant in terms of the roots of the polynomials involved. It can also be obtained by relating $R(f,h)$ to the determinant of multiplication by $f$ on the space $K[x]/(h)$. But is there a way to prove the above formula by extending the three Sylvester matrices involved by cleverly chosen blocks so that the determinants don't change, the three matrices become square matrices of equal size, and one matrix is the product of the other two?
2026-02-23 09:03:42.1771837422
Alternative proof for multiplicativity of resultant?
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