Let ${\mathbb K} \subseteq {\mathbb C}$ be a finite extension of $\mathbb Q$, and let $n=[{\mathbb K} : {\mathbb Q}]$. Let $X_{\mathbb K}$ denote the set of all “components” (i.e., real and imaginary parts) of elements in ${\mathbb K}$. Let ${\mathbb L}=\mathbb Q(X_{\mathbb K})$ ; this is a subfield of $\mathbb R$.
It is easy enough to see that $[{\mathbb L}:{\mathbb Q}]$ is finite, and in fact, it is $\leq 2n^2$ (see proof below). Denote by $f(n)$ the best possible bound ; I have computed that $f(2)=2$ and $f(3)=6$, and I ask : what is $f(n)$ in general ?
Proof of the upper bound
Let ${\mathbb M}_1$ denote the image of $\mathbb K$ by complex conjugation, ${\mathbb M}_2$ the smallest subfield of $\mathbb C$ containing both $\mathbb K$ and ${\mathbb M}_1$ (the so-called “compositum”), and finally ${\mathbb M}_3={\mathbb M}_2[i]$.
By construction, ${\mathbb M}_3$ contains $i$ and is invariant by complex conjugation, so it is also invariant with respect to taking real and imaginary parts. Since it contains $\mathbb K$, we see that ${\mathbb L} \subseteq {\mathbb M}_3$.
Now $[{\mathbb M}_2:{\mathbb Q}]$ is smaller than the product $[{\mathbb K}:{\mathbb Q}][{\mathbb M}_1:{\mathbb Q}]=n^2$, and in turn $[{\mathbb M}_3:{\mathbb Q}]$ is smaller than $2[{\mathbb M}_2:{\mathbb Q}] $, qed.
Update at 17:39 (in answer to Qiaochu’s comment) : it is not true that $f(n)=2n$ for every $n \geq 3$. Indeed, I can show that $f(4) \geq 12$. To check this, let $P$ be any rational polynomial of degree $4$ with no real roots, no purely imaginary root, and Galois group $S_4$ (for example, $P=X^4 - 6X^3 + 15X^2 - 19X + 13$ will do). Then take ${\mathbb K}={\mathbb Q}(\lambda)$ where $\lambda$ is any root of $P$. Using Galois theory, it is easy to see that in this case $[{\mathbb L}:{\mathbb Q}]$ is 12.
Another way to put it : under those hypotheses, there is an automorphism $\sigma$ of $\mathbb C$ fixing $i$ and $\lambda$ but not $\bar{\lambda}$. This $\sigma$ acts as the identity on $\mathbb L$, but acts non-trivially on the real part of ${\mathbb K}[i]$.