Transitivity of Algebraic Extensions

Theorem
Let $E/F/K$ be a tower of field extensions.

If $E$ is algebraic over $F$ and $F$ is algebraic over $K$, then $E$ is algebraic over $K$.

Proof
Let $x \in E$.

There are $a_0,\ldots,a_n \in F$ such that $a_0 + \cdots + a_n x^n = 0$.

Let $L = K(a_0,\ldots,a_n)$.

We have that $L/K$ is finitely generated and algebraic.

Therefore by Finitely Generated Algebraic Extension is Finite this extension is finite.

Since $L(x)/L$ is simple and algebraic, by Structure of Algebraic Field Extension, this extension is also finite.

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

That is, $x$ is contained in a finite extension of $K$.

Therefore because a finite extension is algebraic, it follows that $x$ is algebraic over $K$, as was to be proved.