Center of Division Ring is Subfield
Jump to navigation
Jump to search
Theorem
Let $\struct {K, +, \circ}$ be an division ring.
Let $\map Z K$ be the center of $K$.
Then $\map Z K$ is a subfield of $K$.
Proof
For $\map Z K$ 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
- 1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text {IV}$: Rings and Fields: $24$. The Division Algorithm: Theorem $24.9$