Let $q$ be a prime power, and $D$ a non-commutative division algebra (skew field) over $\mathbb{F}\_q$ (the finite field with $q$ elements) such that the center $C(D)$ equals $\mathbb{F}\_q$.



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> Question 1: Do such division algebras exist? Are there "simple" examples?
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The only non-commutative division algebras over finite fields I know are skew Laurent series. E.g. $D=\mathbb F\_{q^2}((t))$, where $t$ induces by conjugation on the coefficients $c\in\mathbb F\_{q^2}$ the unique involutory field automorphism, i.e. $tc = c^qt$.


Then $\mathbb F\_q$ embeds into $D$, but $C(D)$ is infinite as it contains all even powers of $t$. Maybe the construction can be modified to make the center finite? I am not quite sure where to look for more information or examples. All literature I found so far deals primarily with finite-dimensional division algebras.



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> Question 2: What are good references on infinite dimensional division algebras? Books, articles, surveys, anything?
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Based on an example for such an object, I hope to get a better feeling for the following problem: We now also have an automorphism $\gamma$ of $D$ of order 1 or 2, and an involutory anti-automorphism $\sigma$ of $D$.


Call a (commutative!) subfield $\mathbb K$ of $D$ *nice* if $\gamma$ and $\sigma$ restrict to it, i.e. if $\mathbb K^\sigma = \mathbb K = \mathbb K^\gamma$.



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> Question 3: What can be said about the nice subfields of $D$? In particular, how large can one make nice subfields; are there infinite ones?
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Clearly the (finite) center is nice. And I think that one can always find a nice subfield with $q^2$ elements, but beyond that... ? Maybe looking at some special cases helps:



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> Question 4: What can we say if $\gamma$ is trivial?
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If $D$ contains an element $t$ of infinite (multiplicative) order, then we get an infinite nice subfield: Either $t+t^\sigma$ has infinite order and then we can adjoin that to the center; or else, let $n$ be its finite order. Then $1=(t+t^\sigma)^{np}=t^{np}+(t^{np})^\sigma$, hence $t^{np}$ (which has infinite order) commutes with $(t^{np})^\sigma$ can we can adjoin that to the center. This motivates 



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> Question 5: Does $D$ necessarily contain an element of infinite (multiplicative) order?
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Of course one can now look at further special cases of my problem, e.g. the case that $\gamma$ and $\sigma$ commute seems to be in reach. But this still seems pretty far from the general question.



Infinite dimensional division algebras with finite center, and their involutions
PM Cohn's "Skew fields: Theory of general divison rings", CUP 1995 should prove a valuable reference. 


In particular, Proposition 2.3.5 states that for each field $k$ there is a skew field $D$ whose centre is $k$ and such that $D$ is infinite-dimensional over $k$.


