What is the smallest field which contains all square roots of positive rational numbers? I guess I mean “smallest” in terms of set inclusion, i.e. the minimal one with regard to the “$\subseteq$” relation. The smallest field I know about would be the real algebraic numbers, but I guess that by restricting the degree of minimal polynomials, a smaller field might be possible.
To express this as a formula, I'm looking for the field generated by the set $$ S = \left\{\pm\sqrt x\;\middle\vert\;x\in\mathbb Q\right\} $$
If you know about a field smaller than the algebraic reals which contains $S$, I'd like to know its name and its structure. If you have an argument why the algebraic reals are the smallest field containing $S$, then I'd like to hear the argument or a reference to it.
Update: Answers below indicate that there is such a field containing $S$ and smaller than the real algebraic numbers. So the main issue is finding an established name for this, if there is one.