  The notion of symmetry in polynomial rings with several indeterminates is
generalized to polynomial rings over finite fields. Families of extensions of
the projective line over a finite field of constants possessing this property
are explicitly constructed. These extensions exhibit complete splitting of all
finite rational places. Subcovers of these extensions are also explicitly
described. New examples of function fields attaining the Oesterle bounds are
obtained. These constructions are compared with class field theoretic
constructions achieving similar splitting of rational places. A generalization
to the Hermitian function field over fields of non-square cardinality is
proposed.
