I'm reading Silverman's book the Arithmetic of Elliptic Curves. I don't know how to prove part c of proposition I.2.6 on page 10.
Proposition 2.6. (c) If an affine variety $V$ is defined over $K$, then $\overline{V}$ is also defined over $K$. (Here $\overline{V}$ is the projective closure of $V$.)
What I tried so far: I know that $I(V)=(S)$ for some $S$ containing polynomials over $K$. But by definition of $\overline{V}$, $I(\overline{V})=(f^{*}:f\in I(V))$, where $f^{*}$ is the homogenization of $f$ with respect to $X_{i}$. I'm guessing that $I(\overline{V})=(f^{*}:f\in S)$. So I'm trying to prove $(f^{*}:f\in I(V))=(f^{*}:f\in S)$. Let $(f_{1}g_{1}+\cdots+f_{m}g_{m})^{*}\in LHS$, where $f_{i}\in S$. I cannot move the * inside. I have to multiply by some powers of $X_{i}$. So I don't know what to do next.