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 $$K$$.