Wedderburn's Little Theorem says that every finite Division Ring is commutative. What is about an infinite Division Ring with prime characteristics? Is this also a Field?
[Math] Example of a Non-Commutative Division Ring With Finite Characteristics
abstract-algebraring-theory
Best Answer
Let $F$ be any field with a non identity automorphism $\sigma$ and finite characteristic. (You could, for example, use the Frobenius endomorphism of a nonperfect field.)
The twisted polynomial ring $F[x;\sigma]$ is the set of polynomials written with coefficients on the left of powers of $x$, and the multiplication dictated by $xa=\sigma(a)x$.
Since $\sigma$ isn't the identity map, this yields a noncommutative Noetherian domain. This being the case, it has a "division ring of quotients" which must share the same finite characteristic with $F$ and $F[x;\sigma]$. Clearly it is also infinite.