Exercise 18, § 3

Let K be a field and $f = g/h$ an element of the field of rational functions $K(U, V)$ in two indeterminates, where g and h are two relatively prime polynomials of $K[U, V]$. Show that in the field of rational functions

in $2n$ indeterminates the determinant $\det(f(X_i, Y_j))$ is equal to: where F is a polynomial in $K[X_1, Y_1, \ldots, X_n, Y_n]$ and $V(X_1, \ldots, X_n)$ is the Vandermonde determinant (Algebra, Chapter 111, § 6, no. 4). Consider the particular case where $f = 1/(U + V)$ ("Cuuchy's identity").