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 $\mathbb{F}_q$.

Question 1: Do such division algebras exist? Are there "simple" examples?

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.

Question 2: What are good references on infinite dimensional division algebras? Books, articles, surveys, anything?

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 $\mathbb K^\sigma = \mathbb K = \mathbb K^\gamma$.

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?

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:

Question 4: What can we say if $\gamma$ is trivial?

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

Question 5: Does $D$ necessarily contain an element of infinite (multiplicative) order?

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.