Algebraic extension and infinite polynomial ring Let $K$ a field and $F=K(\alpha_i : i\in I)$ an algebraic extension of $K$.
Is it true that for all $z\in F$ there exists $i_1,\dots,i_k\in I$ such that $z\in K(\alpha_{i_1},\dots,\alpha_{i_n})$ ?
Moreover, does $F$ equals to $K[\alpha_i : i\in I]$ ?
These results are true if $I$ is finite, but I don't know if they remain so if $I$ is infinite, since the proof doesn't seem straightforwardly generalisable. I haven't been able to find a book with this result either.

By definition $F=K(\alpha_i : i\in I)$ is such that for all $z\in F$ there exist $i_1,\dots,i_k\in I$ such that $z\in K(\alpha_{i_1},\dots,\alpha_{i_n})$, so the answer to your first question is trivially yes. For such $i_1,\dots,i_k$ one has $K(\alpha_{i_1},\dots,\alpha_{i_n})=K[\alpha_{i_1},\dots,\alpha_{i_n}]$ so $F$ is also the smallest ring containg all these rings, which is by definition $K[\alpha_i : i\in I]$, so the answer to the second question is yes as well.
