Example 10.141.12. Let $n , m \geq 1$ be integers. Consider the ring map

of Example 10.134.7. Write symbolically

where for example $a_1(b_ i, c_ j) = b_1 + c_1$. The matrix of partial derivatives is

The determinant $\Delta $ of this matrix is better known as the *resultant* of the polynomials $g = x^ n + b_1 x^{n - 1} + \ldots + b_ n$ and $h = x^ m + c_1 x^{m - 1} + \ldots + c_ m$, and the matrix above is known as the *Sylvester matrix* associated to $g, h$. In a formula $\Delta = \text{Res}_ x(g, h)$. The Sylvester matrix is the transpose of the matrix of the linear map

Let $\mathfrak q \subset S$ be any prime. By the above the following are equivalent:

$R \to S$ is étale at $\mathfrak q$,

$\Delta = \text{Res}_ x(g, h) \not\in \mathfrak q$,

the images $\overline{g}, \overline{h} \in \kappa (\mathfrak q)[x]$ of the polynomials $g, h$ are relatively prime in $\kappa (\mathfrak q)[x]$.

The equivalence of (2) and (3) holds because the image of the Sylvester matrix in $\text{Mat}(n + m, \kappa (\mathfrak q))$ has a kernel if and only if the polynomials $\overline{g}, \overline{h}$ have a factor in common. We conclude that the ring map

is étale.

## Comments (2)

Comment #3967 by Manuel Hoff on

Comment #4102 by Johan on