Division rings over fields

Question

Given a field $K$. Rings are assumed to be with identity and associative.

Question 1: Is there a (easy) construction of a non-commutative division ring with center $K$?

Question 2: Is there a complete classification of non-commutative division rings with center $\mathbb{R}$?

Answer 1

Question 1: It depends what you mean by "easy".

There's a fairly easy construction that works if $K$ has characteristic zero:

Let $K(t)$ be the field of rational functions over $K$, let $\sigma$ be the $K$-automorphism of $K(t)$ that sends $t$ to $2t$, and let $D=K(t)((x;\sigma))$ be the twisted Laurent series ring.

I.e., elements of $D$ are Laurent power series $$a_{-n}(t)x^{-n}+\dots+a_{-1}(t)x^{-1}+a_0(t)+a_1(t)x+\dots,$$ where $a_i(t)\in K(t)$, with multiplication twisted by the rule $xt=2tx$.

Then $D$ is a division ring with centre $K$. (Actually, replacing $2$ by any element of $K$ with infinite multiplicative order, this also works for fields of prime characteristic that are not algebraic over the prime subfield.)

Chapter 5 of Lam's "First Course in Noncommutative Ring Theory" describes other constructions (or look at Cohn's book on "Skew Fields" for a much more extensive treatment). I think the "Mal'cev-Neumann construction" gives a division ring with centre $K$ for an arbitrary field $K$.

Question 2: I think it's extremely unlikely that a classification is feasible.