Exercise 18, § 3

Let $K$ be a field, $f = g/h$ an element of the field of rational fractions $K(U, V)$ in two indeterminates, where $g$ and $h$ are two relatively prime polynomials of $K[U, V]$. Now prove that in the field of rational fractions $K(X_1, Y_1, \ldots, X_n, Y_n)$ in $2n$ indeterminates, the determinant $\det(f(X_i, Y_j))$ is equal to:

where $F$ is a polynomial of $K[X_1, Y_1, \ldots, X_n, Y_n]$ and $V(X_1, \ldots, X_n)$ is the Vandermonde determinant (Alg., chap. III, § 6, No. 4). Special case where $f = 1/(U + V)$ (“Cauchy identity”).