Let $F\subset K$ be a field extension and $\alpha_1,\dots,\alpha_n\in F$ algebraic elements over $K$. Is it true that
$K(\alpha_1,\dots,\alpha_n)=K[\alpha_1,\dots,\alpha_n]\ \text{?}$
Indeed, $\alpha_1$ is algebraic over $K$, thus $K(\alpha_1)=K[\alpha_1]$. Then, $\alpha_2$ is algebraic over $K(\alpha_1)\subset K$, thus $K(\alpha_1,\alpha_2)=K(\alpha_1)(\alpha_2)=K(\alpha_1)[\alpha_2]=K[\alpha_1][\alpha_2]=K[\alpha_1,\alpha_2]$. By induction, we get the result.
Is there a flaw ? I don't feel sure with this.