Let $F$ an extension field of $K$ and $X$ a subset of algebraic elements of $F$ over $K$, $K(X)$ the intersections of all fields containing $K$ and $X$, and $K[X]$ the intersections of all rings containing $K$ and $X$.
Is it $K(X)=K[X]$?
In case $X$ is finite, I can prove it with induction starting from $K(a)=K[a]$ with $a$ algebraic (this result is in textbooks) then my question is for X infinite.
Yes, because every element of $K(X)$ belongs to $K(Y)$ where $Y$ is a finite subset of $X$: $$ K(X) = \bigcup_{\substack{Y \subseteq X \\ Y \text{ finite}}} K(Y) = \bigcup_{\substack{Y \subseteq X \\ Y \text{ finite}}} K[Y] = K[X] $$