Notation in Silverman's Arithmetic of EC - ideals

Question

I have a question about the notation in Silverman's Arithmetic of EC. On page 2, "Remark 1.2 Let V be an algebraic set, and consider the ideal $I(V/K)$ defined by $$I(V/K)=\{f\in K[X]:f(P)=0\text{ for all } p\in V\}=I(V)\cap K[X].$$ Then we see that $V$ is defined over $K$ if and only if $$I(V)=I(V/K)\overline{K}[X]."$$ What does $I(V/K)\overline{K}[X]$ mean?