  Let $d$ be a positive integer and $\mathbb H$ be an integrally closed subring
of a global function field $F$. The purpose of this paper is to provide a
general sieve method to compute densities of subsets of $\mathbb H^d$ defined
by local conditions. The main advantage of the method relies on the fact that
one can use results from measure theory to extract density results over
$\mathbb H^{d}$. Using this method we are able to compute the density of the
set of polynomials with coefficients in $\mathbb H$ which give rise to "good"
totally ramified extensions of the global function field $F$. As another
application, we give a closed expression for the density of rectangular
unimodular matrices with coefficients in $\mathbb H$ in terms of the
$L$-polynomial of the function field.
