I want to know why is my idea wrong in theorem

$F(u_1,\cdots,u_n)$ is generated by $1$ and $u_1,\cdots,u_n$ as a field extension of $F$ (in the in algebraic case, as a $F$-algebra as well), but not as a $F$-vector space, and here you mean dimension as $F$-vector space. If the extension is algebraic, then $F(u_1,\cdots,u_n)$ is generated as a $F$-vector space by the monomials $u_1^{\alpha_1}u_2^{\alpha_2}\cdots u_n^{\alpha_n}$, where $(\alpha_1,\cdots,\alpha_n)\in\Bbb N^n$.

Consider the $F=\Bbb{Q}$ and $f(x)=x^3-2$ and find its splitting field $K$ over $F$. You will get $K=\Bbb{Q}(\sqrt[3]{2},\zeta_3)$, so $[K:F]=6>3+1$. Here you try to get where is the problem in your approach.