Maximal Algebraic Extension is Subfield

Theorem

Let $L / K$ be a field extension.

Let $K^a$ be the maximal algebraic extension of $K$ contained in $L$.

Then $K^a$ is a subfield of $L$.

Proof

Let $\alpha, \beta \in K^a$.

By Field Adjoined Algebraic Elements is Algebraic, $\map K {\alpha, \beta} / K$ is algebraic.

By definition, $\map K {\alpha, \beta}$ is a field.

Therefore $\alpha \beta$, $\alpha^{-1}$ and $\alpha - \beta$ all lie in $\map K {\alpha, \beta}$.

Hence all are algebraic over $K$.

Also: $K \subseteq K^a$ so:

$K^a \ne \O$

By the Subfield Test:

$K^a \le L$ is a subfield of $L$.

$\blacksquare$