Given a prime number $p$, can you give me concrete examples of fields $\mathbf F$ of characteristic $p$ and quaternion algebras $\mathbb H(\mathbf F)$ over $\mathbf F$ such that $\mathbb H(\mathbf F)$ is a (non-commutative) division ring ?



Quaternion algebra in characteristic $p$
If $F$ is a finite field, the unique central division $F$-algebra is $F$ itself. This is Wedderburn's theorem :


<https://en.wikipedia.org/wiki/Wedderburn%27s_little_theorem>


In particular there is no central quaternion $F$-algebra. In contrast there are central division $F$-algebras of arbitrary (square) dimension when e.g. $F$ is a global field (a number field, or a function field over a finite field $F\_{q}$) or a local field (a finite extension of ${\mathbb Q}\_p$, or a field of the form $F\_q ((X))$, the field of Laurent series over a finite field $F\_q$). All of this is classical and may be found in any lecture notes on the theory of Brauer group :


<https://en.wikipedia.org/wiki/Brauer_group>


A. Weil, Basic Number Theory


