Intersection of All Division Subrings is Prime Subfield

Theorem
Let $\left({K, +, \circ}\right)$ be a division ring.

Let $P$ be the intersection of the set of all division subrings of $K$.

Then $P$ is the prime subfield of $K$.

Proof

 * By Intersection of Division Subrings, the intersection $P$ of the set of all division subrings of $K$ is a division ring.

Let $Z \left({K}\right)$ be the center of $K$.

From Center of Ring is Commutative Subring, $Z \left({K}\right)$ is a commutative subring of $K$ and therefore a commutative division ring, thus is a subfield of $K$.

But as $P$ is contained in $Z \left({K}\right)$, it is itself commutative.

By its definition, $P$ contains no proper subfield and hence is a prime field.

Also, $P$ is contained in every other subfield of $K$ and is therefore the only prime subfield of $K$.