  Let $M$ be a smooth compact $CR$ manifold of $CR$ dimension $n$ and $CR$
codimension $k$, which has a certain local extension property $E$. In
particular, if $M$ is pseudoconcave, it has property $E$. Then the field $\Cal
K(M)$ of $CR$ meromorphic functions on $M$ has transcendence degree $d$, with
$d\leq n+k$. If $f_1, f_2, \hdots , f_d$ is a maximal set of algebraically
independent $CR$ meromorphic functions on $M$, then $\Cal K(M)$ is a simple
finite algebraic extension of the field $\Bbb C(f_1, f_2, \hdots, f_d)$ of
rational functions of the $f_1, f_2, \hdots , f_d$. When $M$ has a projective
embedding, there is an analogue of Chow's theorem, and $\Cal K(M)$ is
isomorphic to the field $\Cal R(Y)$ of rational functions on an irreducible
projective algebraic variety $Y$, and $M$ has a $CR$ embedding in $\roman{reg}
Y$. The equivalence between algebraic dependence and analytic dependence fails
when condition $E$ is dropped.
