Non commutative field of characteristic $\neq 0$

Question

Does there exist a non-commutative field of a characteristic different from zero?

Answer 1

Usually we call "non-commutative fields" division algebras (the terminology "skew field" also exists, but "division algebra" is really the modern term).

And yes they exist in all characteristic: if $K$ is any field, the so-called Brauer group $\operatorname{Br}(K)$ gives the set of isomorphism classes of finite-dimensional central division $K$-algebras. If $K$ has characteristic $p$, then so do those algebras.

You might have asked this question because if $K$ is finite then $\operatorname{Br}(K)$ is trivial, so our favorite fields of characteristic $p$ (the finite fields) do not yield non-commutative division algebras.

But in general $\operatorname{Br}(K)$ is far from trivial. For instance, $\operatorname{Br}(\mathbb{F}_p(X))$ has a nice description in terms of the points of the projective line $\mathbb{P}^1(\mathbb{F}_p)$.

Here is a concrete example of a division algebra in characteristic $p\neq 2$: define $Q$ as an $\mathbb{F}_p(X,Y)$-algebra generated by two elements $i$ and $j$ such that $i^2=X$, $j^2=Y$ and $ij=-ji$. (When $p=2$ this does not work, you have to replace by $i^2+i=X$, $j^2=Y$ and $ji=ij+j$.) It is a so-called quaternion algebra over $\mathbb{F}_p(X,Y)$, of dimension $4$.