Transitivity of Algebraic Extensions

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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 $L = K \left({a_0, \ldots, a_n}\right)$.

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 $L \left({x}\right) / L$ is simple and algebraic.

So by Structure of Simple Algebraic Field Extension, this extension is also finite.

Therefore, by the Tower Law:

$L \left({x}\right) / K$ 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.

$\blacksquare$


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