Given an algebraically closed field $K$ and a zero dimensional ideal $I \subset K[x,y]$ ($1 \notin I$), then there exists a (univariate) polynomial $p \in I \cap K[x]$, s.t. $I \cap K[x] = \left< p\right>$. A necessary condition for a common root $(x_i,y_i)$ of $I$ (i.e. $(x_i,y_i) \in V(I)$) is then, of course, that $x_i$ is a root of $p$.
But what about the reverse direction: can any root $\bar x$ of $p$ be extended to a common root $(\bar x,\bar y)$ of $I$?
It seems obvious that this is true, but I cannot find a proof.