This is a question about Remark 1.2 of Silverman's "Arithmetic of Elliptic Curves".
Given a perfect field $K$, it's stated that an algebraic set $V$ is defined over $K$ if and only if $$I(V)=I(V/K)\overline{K}[X].$$ Here $I(V)$ is the ideal of $V$, $I(V/K)=I(V) \cap K[X]$ and $\overline{K}$ is an algebraic closure of $K$. I'm not sure how to understand the second part of this equivalence. Does it mean that an arbitrary $f \in I(V)$ is of the form $gh,$ with $g \in I(V/K)$ and $h \in \overline{K}[X]$? If not, what?
I think the points is that $I(V|K)=(f_1,\dots,f_n)$ is generated by polynomials $f_i \in K[X]$, so their coefficients are in $K$ but you consider $I(V)$ as an ideal of $\bar{K}[X]$, so write $I(V)=I(V|K)\bar{K}[X]$ hence every element of $I(V)$ is a linear combination $\sum_ig_if_i$ where $g_i \in \bar{K}[X]$