Subalgebra of Algebraic Field Extension is Field
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Theorem
Let $E / F$ be an algebraic field extension.
Let $A \subseteq E$ be a unital subalgebra over $F$.
Then $A$ is a field.
Proof
By Integral Ring Extension is Integral over Intermediate Ring, $E$ is integral over $A$.
Let $a \in A$ be nonzero.
Because $E$ is a field, $a$ is a unit of $E$.
By Ring Element is Unit iff Unit in Integral Extension, $a$ is a unit of $A$.
Thus $A$ is a field.
$\blacksquare$