The algebraic elements of $\mathbb{R}$ are those elements which are roots of nonzero polynomials with coefficients in $\mathbb{Q}$. In fact, by multiplying through by denominators, we can even take the polynomials to have coefficients in $\mathbb{Z}$. My question is, in a sense, about the converse of this situtation in a more general setting.
Let $\mathbb{L}$ be a field extension of a field $\mathbb{K}$, and let $\mathbb{S}$ be a subring of $\mathbb{K}$ such that every algebraic element of $\mathbb{L}$ is the root of some nonzero polynomial with coefficients in $\mathbb{S}$. Is $\mathbb{K}$ the field of quotients of $\mathbb{S}$?
Note, as $\mathbb{S}$ is a subring of the field $\mathbb{K}$, it is automatically an integral domain so we can construct the field of quotients of $\mathbb{S}$.
As Mariano and pritam's examples show, the answer is no. However, in all of these examples, $\mathbb{K}$ is an algebraic extension of the field of quotients of $\mathbb{S}$. Is this always the case?