Maximal Algebraic Extension is Subfield

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Theorem

Let $L/K$ be a extension of fields.

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, $K(\alpha,\beta)/K$ is algebraic.

By definition $K(\alpha,\beta)$ is a field.

Therefore $\alpha\beta$, $\alpha^{-1}$ and $\alpha - \beta$ all lie in $K(\alpha,\beta)$, hence are algebraic over $K$.

Also $K \subseteq K^a$ so $K^a \neq \emptyset$, and by the Subfield Test, $K^a \leq L$ is a subfield.

$\blacksquare$