Finitely Generated Algebraic Extension is Finite

Theorem
Let $L/K$ be a field extension and $\alpha_1,\ldots,\alpha_n \in L$ algebraic over $K$.

Then $K(\alpha_1,\ldots, \alpha_n)/K$ is finite.

Proof
Let $S = \{\alpha_1,\ldots,\alpha_n\}$.

We show by induction on $n$ that $K(S)/K$ is finite.

Clearly $K$ is finite over itself, so the result holds when $n = 0$.

Now suppose that for all sets $T \subseteq L$ with $|T| \leq n-1$ and each element of $T$ algebraic over $K$, $K(T)/K$ is finite. We have:


 * $K(S) = K(\alpha_1,\ldots, \alpha_{n-1})(\alpha_n)$

By the induction hypothesis $K(\alpha_1,\ldots, \alpha_{n-1})/K$ is finite.

By the structure of a simple algebraic extension $K(S)/K(\alpha_1,\ldots, \alpha_{n-1})$ is finite.

Therefore, by the tower law, $K(S) / K$ is finite.