  A famous theorem of Hilbert from 1888 states that a positive semidefinite
(psd) real form is a sum of squares (sos) of real forms if and only if $n=2$ or
$d=1$ or $(n,2d)=(3,4)$, where $n$ is the number of variables and $2d$ the
degree of the form. In 1976, Choi and Lam proved the analogue of Hilbert's
Theorem for symmetric forms by assuming the existence of psd not sos symmetric
$n$-ary quartics for $n \geq 5$. In this paper we complete their proof by
constructing explicit psd not sos symmetric $n$-ary quartics for $n \geq 5$.
