Transitivity of Algebraic Extensions
Theorem
Let $E / F / K$ be a tower of field extensions.
Let $E$ be algebraic over $F$.
Let $F$ be 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 $L = \map K {a_0, \ldots, a_n}$.
We have that $L / K$ is finitely generated and algebraic.
Therefore by Finitely Generated Algebraic Extension is Finite this extension is finite.
We have that $\map L x / L$ is simple and algebraic.
So by Structure of Simple Algebraic Field Extension, this extension is also finite.
Therefore, by the Tower Law:
- $\map L x / K$ is finite.
That is, $x$ is contained in a finite extension of $K$.
Therefore because a Finite Field Extension is Algebraic, it follows that $x$ is algebraic over $K$, as was to be proved.
$\blacksquare$