Hilbert's Nullstellansatz

Let $K$ be an algebraically closed field and let $I$ be an Ideal in $K(x)$, where $x=(x_1, x_2, \ldots, x_n)$ is a finite set of indeterminates. Let $p \in K(x)$ be such that for any $(c_1, \ldots, c_n)$ in $K^n$, if every element of $I$ vanishes when evaluated if we set each ($x_i=c_i$), then $p$ also vanishes. Then $p^j$ lies in $I$ for some $j$. Colloquially, the theory of algebraically closed fields is a complete model.