# How to construction a division ring from the given field?

Let $F$ be a field of characteristic 2. How could we construct a division ring $D$ which centre is $F$.

Where division ring mean non-commutative ring with unity $1$ and for each non-zero element $x \in D$ there exists $x^{-1} \in D$ such that $xx^{-1}=x^{-1}x=1$.

• What about taking $D$ a quaternion algebra over $\mathbb{F}_2$? – Dietrich Burde Apr 9 '18 at 18:53
• I think that your "quaternionic" algebra will be (disappointingly) commutative by little Wedderburn. – xsnl Apr 9 '18 at 19:25