Can an algebraic variety be defined over a larger field?

Question

Let $K$ and $L$ be a field which satisfies $K⊂L$. If affine algebraic variety $E$ is defined over some field $K$, is $E$ defined over $L$ ?

I think this is obvious because it's rational point does not change by the action of $Gal( \overline{K}/K)$, it also does not change under the action of $Gal( \overline{K}/L)$. But I heard following kind of argument in a cession.

A: This curve $E$ is defined over $ \Bbb{C}$, ・・・

B: Is your curve $E$ is appriori defined over $ \Bbb{Q}(i)$ ?

So I'm confused.

Answer 1

A scheme $X\to\operatorname{Spec} k$ over a field $k$ is defined over a subfield $\ell\subset k$ iff there is a scheme $X_0\to\operatorname{Spec}\ell$ such that $X_0\times_{\operatorname{Spec} \ell} \operatorname{Spec} k \to \operatorname{Spec} k$ is isomorphic to $X\to\operatorname{Spec} k$. So the answer to your question is yes.