Intersection of All Division Subrings is Prime Subfield

Theorem

Let $\struct {K, +, \circ}$ 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 is Division Subring, the intersection $P$ of the set of all division subrings of $K$ is a division ring.

Let $\map Z K$ be the center of $K$.

From Center of Ring is Commutative Subring, $\map Z K$ is a commutative subring of $K$.

Therefore $\map Z K$ is a commutative division ring

Thus $\map Z K$ is a subfield of $K$.

But as $P$ is contained in $\map Z K$, 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 $K$.

Therefore $P$ is the only prime subfield of $K$.

$\blacksquare$