In Milne's field theory notes he defines, given an extension $E/F$ and a subset $S\subset E$, the subfield of $E$ generated by $F$ and $S$ as the smallest subfield of $E$ containing both $F$ and $S$.
Given again a field extension $E/F$ and a subset $S\subset E$, he defines the subring of $E$ generated by $F$ and $S$ as the smallest subring of $E$ containing both.
First of all, I'd like to understand if this second definition can really be written without any reference to $E$ by just saying it's the polynomial algebra $F[S]$ in the usual way.
Second, I would like to understand exactly why we must mention $E$ in the subfield case. I know there are no free fields, but can someone explain precisely why we need an existing extension $E/F$ to say anything? Could I have defined the generated subfield by $F$ and just some set $S$ as a minimal extension $E/F$ containing $S$?
In either case you can certainly define the minimal ring/field containing $F$ and $S$, but if $S$ is not given as a subset of a pre-fixed extension $E$ you need to specify what algebraic relations, if any, do the elements in $S$ satisfy.
When taking $S\subset E$, the relations satisfied by elements in $S$ are somewhat implicit.
For instance, if $K=\Bbb Q$ and $E=\Bbb C$ the subextension $\Bbb Q(\sqrt{2})$ and $\Bbb Q(\pi)$ are not isomorphic even though they are obtained by adding just one element.