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 ?
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4$\begingroup$ Take $\mathbf{F}=\mathbb{F}_p(X,Y)$ and $\mathbb{H}(F)$ the algebra with basis $1,i,j,k$ and relations $ij=-ji=k$, $i^2=X$, $j^2=Y$. $\endgroup$– abxCommented Feb 3, 2016 at 8:16
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$\begingroup$ @abx, not when $p=2$... $\endgroup$– KConradCommented Feb 3, 2016 at 9:32
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1 Answer
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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
answered Feb 3, 2016 at 8:59