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.


1 Answer 1


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 $k$.

  • $\begingroup$ Thanks Guntram. Our library doesn't have that book, and Google Books skips exactly over 2.3.5 ... but I managed to finally locate the proposition with help of Amazon's book preview, and it answers my first question quite nicely! $\endgroup$
    – Max Horn
    Dec 3, 2010 at 18:41

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge that you have read and understand our privacy policy and code of conduct.

Not the answer you're looking for? Browse other questions tagged or ask your own question.