Subalgebra of Algebraic 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
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 invertible in $E$.

By Ring Element is Invertible iff Invertible in Integral Extension, $a$ is invertible in $A$.

Thus $A$ is a field.

Weaker statement

 * Subalgebra of Finite Field Extension is Field