Transitivity of Algebraic Extensions

Let $K\subseteq E\subseteq F$ be a tower of fields. If $F/E$ and $E/K$ are algebraic, then $F/K$ is algebraic.

Proof:

(Hungerford pg 237, reworded)

Let $u\in F$. Since $u$ is algebraic over $E$, there exists some $b_i\in E$ ($b_n\neq 0$) such that $\displaystyle b_0+b_1u+\dots+b_nu^n=0.$

Let $L=K(b_0,\dots,b_n)$, then $u$ is algebraic over $L$. Hence $L(u)/L$ is finite. Note that $L/K$ is finitely generated and algebraic, since each $b_i\in E$ is algebraic over $K$. Thus $L/K$ is finite.

Thus by Tower Law, $L(u)/K$ is finite, thus algebraic.

Hence $u\in L(u)$ is algebraic over $K$. Since $u$ was arbitrary, $F$ is algebraic over $K$.