# Division rings over fields

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 $$\mathbb{R}$$?

• For the Problem 2, the only finite dimensional non-commutative division ring is the quaternions $\mathbb{H}$. en.wikipedia.org/wiki/… Apr 25, 2021 at 11:10
• @Muses_China Even in infinite dimensions?
– Mare
Apr 25, 2021 at 11:11

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.