Subalgebra of Finite Field Extension is Field

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.

Proof 2
By Finite Field Extension is Algebraic, $E/F$ is algebraic.

The result follows from Subalgebra of Algebraic Field Extension is Field