  We introduce the notion of "quasi-symmetric" polynomials, which is a
generalization of the notion of symmetry, and is particularly suited to the
setting of polynomial rings over finite fields. The properties of this new
class of functions are studied. It is then shown that quasi-symmetric
polynomials are very useful in splitting rational places in extensions of
function fields over finite fields and thus may be used in constructing
extensions with many rational places.
