In this paper we first prove a theorem on the
nonexistence of pyramidal polynomial basis functions. Then we
present a new symmetric composite pyramidal finite element which
yields a better convergence than the nonsymmetric one. It has
fourteen degrees of freedom and its basis functions are incomplete
piecewise triquadratic polynomials. The space of ansatz functions
contains all quadratic functions on each of four subtetrahedra that
form a given pyramidal element.