Center of Division Ring is Subfield

Theorem

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

Then $Z \left({K}\right)$, the center of $K$, is a subfield of $K$.

Proof

For $Z \left({K}\right)$ to be a subfield of $K$, it needs to be a division ring that is commutative.

Thus the result follows directly from Center of Ring is Commutative Subring.

$\blacksquare$

Sources

• Seth Warner: Modern Algebra (1965)... (previous)... (next): $\S 24$: Theorem $24.9$