What is a "best" transcendence basis for R/Q ?

Question

It is easy to show, using the axiom of Zorn, that there exists a transcendence basis for $\mathbb{R}/\mathbb{Q}$, i.e. a set $S$, algebraically independent over $\mathbb{Q}$, such that $\mathbb{R}/\mathbb{Q}(S)$ is an algebraic extension.

What can we say about $T=\mathbb{R} - \mathbb{Q}(S)$? It is easy to show that $\mathbb{R}/\mathbb{Q}$ is not purely transcendental, so that $T \neq \emptyset$. (Indeed, no element of $\mathbb{R}-\mathbb{Q}$ algebraic over $\mathbb{Q}$ may be contained in $\mathbb{Q}(S)$ - an algebraic relation for such an element over $\mathbb{Q}$ would immediately translate into a relation of algebraic dependence in $S$ over $\mathbb{Q}$.)

Thus, all real, non-rational algebraic numbers are contained in $T$. Is it possible, for example, to choose $S$ so that this inclusion is an equality? If not, how "small" can we make $T$?

I'm sorry if this turns out to be trivial, or if the answer to my question can be easily found in the literature. I tried!

Thank you very much and have a pleasing day.

Answer 1

No, you cannot choose $T$ so nicely. For cardinality reasons, we can choose some $s\in S$ with $s\notin \overline{\mathbb{Q}}$. Now, consider the element $\sqrt{s}$. I'll leave it to you to consider how large this makes $T$.