Maximal Algebraic Extension is Subfield
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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 $L$.
$\blacksquare$