Let $w \in F(u_{1},u_{2}, \cdots, u_{n})$ where all $u_{i}$ are algebraic over a field $F$. Let $d_{i}$ be the degree of $u_{i}$ over $F$. I need to prove that the degree of $w$ over $F$ is less than or equal to $d_{1}d_{2} \cdots d_{n}$.
To do this, first, I defined $K:=F(u_{1}, u_{2}, \cdots , u_{n})$ where all $u_{i}$ are algebraic over the field $F$.
Then, I obtained $K$ from the following chain of extensions: $$ F \subseteq F(u_{1}) \subseteq F(u_{1}, u_{2}) \subseteq F(u_{1}, u_{2}, u_{3}) \subseteq \cdots F(u_{1},\cdots , u_{n-1}) \subseteq F(u_{1}, \cdots, u_{n}) = K$$
Furthermore, $F(u_{1},u_{2}) = F(u_{1})(u_{2})$, $F(u_{1}, u_{2}, u_{3}) = F(u_{1},u_{2})(u_{3})$, and so we can generalize $F(u_{1},\cdots , u_{i})$ to be the simple extension $F(u_{1},\cdots , u_{i-1})(u_{i})$.
Then, I have a lemma that says that if $L$ is a field such that $F \subseteq K \subseteq L$, and $u \in L$ is algebraic over $F$, then $u$ is algebraic over $K$.
By this lemma, each $u_{i}$ is algebraic over $F$, and hence is algebraic over $F(u_{1},\cdots, u_{i-1})$
And, since every simple extension by an algebraic element is finite dimensional, we have that $[F(u_{1}, \cdots, u_{i}) : F(u_{1}, \cdots , u_{i-1})]$ is finite for each $i = 2, \cdots , n$.
So, we have that $[K:F]=[K:F(u_{1}, \cdots , u_{n-1})] \cdots [F(u_{1},u_{2}, u_{3}): F(u_{1},u_{2})] [F(u_{1},u_{2}):F(u_{1})][F(u_{1}):F]$, and since for each $i = 2,\cdots , n$, $[F(u_{1},u_{2},\cdots , u_{i}): F(u_{1}, u_{2}, \cdots , u_{i-1})] \Large \vert \normalsize [F(u_{1}, u_{2}, \cdots , u_{i}):F] = d_{i}$. Therefore, $[K:F] \leq d_{n} \cdots d_{3} d_{2} d_{1}$
Does what I've said hold water? If not, could somebody please tell me what I need to fix/include/get rid of?
Thank you so much for your time and patience! :)
Consider the following set of elements of $K$: $\{1,u_1,u_1^2,\ldots,u_1^{d_1-1},u_2,u_2u_1,u_2u_1^2,\ldots,u_2u_1^{d_1-1}, \ldots,u_2^{d_2-1},u_1u_2^{d_2-1},u_1^2u_2^{d_2-1},\ldots,u_1^{d_1}u_2^{d_2-1},\ldots\}$, i.e. the set of elements of the form $u_1^{i_1}u_2^{i_2}\cdots u_n^{i_n}$ where $0 \leq i_k < d_k$ (it can be considered the set of monomials that can be formed in $K$). It is not hard to prove that they span $K$ as a vector space and their number is exactly $d_1d_2\cdots d_n$.