Sufficient background:

Let $\mathcal{M}=(M,...)$ be an $\mathcal{L}$-structure and $X\subset M$.

**Definition**. $X$ is *large* if there exists a function $f:\mathcal{M}^n \overset {\leq k} \rightarrow \mathcal{M}$ definable in $\mathcal{M}$ such that $f(X^n)=M$ for some $n$, $k$. Otherwise, $X$ is *small*.

Here, $f$ is a kind of multi-valued function which targets some subsets of $M$ of less than or equal to $k$ elements. But this point is not rather of such importance, as in algebraically closed fields we do not need functions to be that "k-valued" (due to a lemma).

My questionis...In $ACF$, when exactly do we have that a subfield is small?

**E.g.** In $(\mathbb{C}, +, \cdot , 0, 1)$, $\mathbb{R}$ is large since there exists a definable function $f:\mathbb{R}^2 \rightarrow \mathbb{C}$ sending $(x, y)$ to $x+iy$ and thus $f(\mathbb{R}^2)=\mathbb{C}$. However, $\mathbb{Q}$ is small due to the general fact(which can easily be seen) that if $|X| < |M|$ and $|M|$ is infinite, then $|X|$ is small.