Subalgebra of Finite 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 1

By Subring of Integral Domain is Integral Domain, $A$ is an integral domain.

By Subspace of Finite Dimensional Vector Space is Finite Dimensional, $A$ is finite dimensional over $F$.

By Finite-Dimensional Integral Domain over Field is Field, $A$ is a field.

$\blacksquare$

Proof 2

By Finite Field Extension is Algebraic, $E/F$ is algebraic.

The result follows from Subalgebra of Algebraic Field Extension is Field

$\blacksquare$